Polynomials
Then
Now
Why?
In Chapter 1, you performed operations on expressions with exponents.
In Chapter 7, you will:
SPACE The Very Large Array is an arrangement of 27 radio antennas in a Y pattern. The data the antennas collect is used by astronomers around the world to study the planets and stars. Astrophysicists use and apply properties of exponents to model the distance and orbit of celestial bodies.
Simplify expressions involving monomials. Use scientific notation. Find degrees of polynomials, write polynomials in standard form, and add, subtract, and multiply polynomials.
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Foldables
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Worksheets
Tennessee Curriculum Standards CLE 3102.3.2
Get Ready for the Chapter 
Diagnose Readiness
1
You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
QuickCheck
QuickReview
Write each expression using exponents. (Lesson 11)
Example 1
1. 4 · 4 · 4 · 4 · 4
Write 5 · 5 · 5 · 5 + x · x · x using exponents.
2. y · y · y
4 factors of 5 is 5 4.
3. 6 · 6
3 factors of x is x 3.
4. 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2
So, 5 · 5 · 5 · 5 + x · x · x = 5 4 + x 3.
5. b · b · b · b · b · b 6. m · m · m · p · p · p · p · p · p 1 _ 1 _ 1 _ 1 _ 7. _ · 1 ·_ · 1 ·_ · 1 ·_ ·1 3
3
3
3
3
3
3
3
w _ w 8. _xy · _xy · _xy · _xy · _ z · z
Example 2
Evaluate each expression. (Lesson 12) 9. 2 3
10. (5) 2 2 13. _
(3)
12. (4) 3
11. 3 3
2
1 14. _
(2)
4
15. SCHOOL The probability of guessing correctly on 5 true1 5 . Express this probability as a fraction false questions is _
(2)
(_7 )
2 Evaluate 5 . 2
2
(_57 ) = _57 2
25 =_ 49
Power of a Quotient Simplify.
without exponents. Find the area or volume of each figure. (Lessons 08 and 09)
Example 3
16.
Find the volume of the figure.
17. 2m 5 cm 7 cm
5 ft
3 cm
18. PHOTOGRAPHY A photo is 4 inches by 6 inches. What is the area of the photo?
5 ft 5 ft
V = wh = 5 · 5 · 5 or 125
Volume of a rectangular prism = 5, w = 5, and h = 5
The volume is 125 cubic feet.
2
Online Option Take an online selfcheck Chapter Readiness Quiz at connectED.mcgrawhill.com. 399
Get Started on the Chapter You will learn several new concepts, skills, and vocabulary terms as you study Chapter 7. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills.
StudyOrganizer
NewVocabulary
Polynomials Make this Foldable to help you organize your Chapter 7 notes about polynomials. Begin with nine sheets of notebook paper.
1
2
3
Arrange the paper into a stack.
Staple along the left side. Starting with the second sheet of paper, cut along the right side to form tabs.
Label the cover sheet “Polynomials” and label each tab with a lesson number.
Poly
nom
71 72 73 74
ials
English
Español
constant
p. 401
constante
monomial
p. 401
monomio
zero exponent
p. 409
cero exponente
negative exponent
p. 410
exponente negativo
order of magnitude
p. 411
orden de magnitud
scientific notation
p. 416
notación científica
binomial
p. 424
binomio
degree of a monomial
p. 424
grado de un monomio
degree of a polynomial p. 424
grado de un polinomio
polynomial
p. 424
polinomio
trinomial
p. 424
trinomio
leading coefficient
p. 425
coeficiente líder
standard form of a polynomial
p. 425
forma estándar de polinomio
FOIL method
p. 448
método foil
quadratic expression
p. 448
expresion cuadrática
ReviewVocabulary base p. 5 base In an expression of the form x n, the base is x. Distributive Property p. 23 Propiedad distributiva For any numbers a, b, and c, a(b + c) = ab + ac and a(b  c) = ab  ac. exponent p. 5 exponente In an expression of the form x n, the exponent is n. It indicates the number of times x is used as a factor.
400  Chapter 7  Polynomials
exponent
x n = x · x · x · x ·…· x n times base
Multiplying Monomials Then
Now
Why?
You performed operations on expressions with exponents.
1 2
Many formulas contain monomials. For example, the formula for the horsepower
(Lesson 11)
NewVocabulary monomial constant
Multiply monomials. Simplify expressions involving monomials.
v 3 . H represents the of a car is H = w _
( 234 )
horsepower produced by the engine, w equals the weight of the car with passengers, and v is the velocity of the car at the end of a quarter of a mile. As the velocity increases, the horsepower increases.
1 Monomials
A monomial is a number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. It has only v 3 one term. In the formula to calculate the horsepower of a car, the term w _ 234 is a monomial.
( )
ab An expression that involves division by a variable, like _ c , is not a monomial. Tennessee Curriculum Standards ✔ 3102.3.4 Simplify expressions using exponent rules including negative exponents and zero exponents. CLE 3102.4.1 Use algebraic reasoning in applications involving geometric formulas and contextual problems.
A constant is a monomial that is a real number. The monomial 3x is an example of a linear expression since the exponent of x is 1. The monomial 2x 2 is a nonlinear expression since the exponent is a positive number other than 1.
Example 1 Identify Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. a. 10
Yes; this is a constant, so it is a monomial.
b. ƒ + 24
No; this expression has addition, so it has more than one term.
c. h 2
Yes; this expression is a product of variables.
d. j
Yes; single variables are monomials.
GuidedPractice 1B. 23abcd 2
1A. x + 5 2
xyz 1C. _
mp 1D. _ n
2
Recall that an expression of the form x n is called a power and represents the result of multiplying x by itself n times. x is the base, and n is the exponent. The word power is also used sometimes to refer to the exponent. 4 factors
exponent 4
3 = 3 · 3 · 3 · 3 = 81 base
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401
By applying the definition of a power, you can find the product of powers. Look for a pattern in the exponents. 2 factors 2
4 factors
4
2 ·2 =2·2·2·2·2·2
3 factors 2 factors 3
2
4 ·4 =4·4·4·4·4
2 + 4 = 6 factors
3 + 2 = 5 factors
These examples demonstrate the property for the product of powers.
KeyConcept Product of Powers Words
To multiply two powers that have the same base, add their exponents.
Symbols
For any real number a and any integers m and p, a m · a p = a m + p.
Examples
b3 · b5 = b3 + 5 or b8
g4 · g6 = g4 + 6 or g10
Example 2 Product of Powers Simplify each expression. a. (6n 3)(2n 7)
(6n 3)(2n 7) = (6 · 2)(n 3 · n 7)
Group the coefficients and the variables.
= (6 · 2)(n 3 + 7) = 12n
StudyTip Coefficients and Powers of 1 A variable with no exponent or coefficient shown can be assumed to have an exponent and coefficient of 1. For example, x = 1x 1.
Product of Powers
10
Simplify.
b. (3pt 3)(p 3t 4)
(3pt 3)(p 3t 4) = (3 · 1)(p · p 3)(t 3 · t 4) = (3 · 1)(p 1 + 3)(t 3 + 4) 4 7
= 3p t
Group the coefficients and the variables. Product of Powers Simplify.
GuidedPractice 2A. (3y 4)(7y 5)
2B. (4rx 2t 3)(6r 5x 2t)
We can use the Product of Powers Property to find the power of a power. In the following examples, look for a pattern in the exponents. 4 factors
(3
2)4
= (3 2)(3 2)(3 2)(3 2) = 32 + 2 + 2 + 2 = 38
3 factors
(r
4) 3
= (r 4)(r 4)(r 4) = r4 + 4 + 4 = r 12
These examples demonstrate the property for the power of a power.
KeyConcept Power of a Power Words
To find the power of a power, multiply the exponents.
Symbols
For any real number a and any integers m and p, (a m ) p = a m · p.
Examples
(b 3) 5 = b 3 · 5 or b 15
402  Lesson 71  Multiplying Monomials
(g 6) 7 = g 6 · 7 or g 42
StudyTip Power Rules If you are unsure about when to multiply the exponents and when to add the exponents, write the expression in expanded form.
Example 3 Power of a Power 2 4 Simplify ⎡⎣(2 3) ⎤⎦ .
⎡⎣(2 3)2⎤⎦ 4 = (2 3 · 2) 4 4 =(2 6) =2
Power of a Power Simplify.
6·4
Power of a Power
= 2 24 or 16,777,216
Simplify.
GuidedPractice Simplify each expression. 3 2 3B. ⎡⎣(3 2) ⎤⎦
2 4 3A. ⎡⎣(2 2) ⎤⎦
We can use the Product of Powers Property and the Power of a Power Property to find the power of a product. In the following examples, look for a pattern in the exponents. 3 factors
3 factors
(tw) 3 = (tw)(tw)(tw) = (t · t · t)(w · w · w) = t 3w 3
(2yz 2) 3 = (2yz 2)(2yz 2)(2yz 2) = (2 · 2 · 2)(y · y · y)(z 2·z 2·z 2) = 2 3y 3z 6 or 8y 3z 6
These examples demonstrate the property for the power of a product.
KeyConcept Power of a Product
MathHistoryLink Albert Einstein (1879–1955) Albert Einstein is perhaps the most wellknown scientist of the 20th century. His formula E = mc 2, where E represents the energy, m is the mass of the material, and c is the speed of light, shows that if mass is accelerated enough, it could be converted into usable energy.
Words
To find the power of a product, find the power of each factor and multiply.
Symbols
For any real numbers a and b and any integer m, (ab) m = a mb m.
Examples
(2xy3)5 = (2)5x5y15 or 32x5y15
Example 4 Power of a Product GEOMETRY Express the area of the circle as a monomial. Area = πr 2 = π(2xy
Formula for the area of a circle 2 2
)
= π(2 2x 2y 4) 2 4
= 4x y π
Replace r with 2xy 2.
2xy 2
Power of a Product Simplify.
The area of the circle is 4x 2y 4π square units.
GuidedPractice 4A. Express the area of a square with sides of length 3xy 2 as a monomial. 4B. Express the area of a triangle with a height of 4a and a base of 5ab 2 as a monomial. connectED.mcgrawhill.com
403
2 Simplify Expressions
We can combine and use these properties to simplify expressions involving monomials.
KeyConcept Simplify Expressions To simplify a monomial expression, write an equivalent expression in which: • each variable base appears exactly once, • there are no powers of powers, and • all fractions are in simplest form.
Example 5 Simplify Expressions
StudyTip
2 Simplify (3xy 4) ⎣⎡(2y)2⎤⎦3.
Simplify When simplifying expressions with multiple grouping symbols, begin at the innermost expression and work outward.
(3xy 4)2⎡⎣(2y)2⎤⎦3 = (3xy 4) 2(2y) 6 2 =(3) 2x 2(y 4) (2) 6y 6 = 9x 2y 8(64)y 6 2
8
= 9(64)x · y · y = 576x 2y 14
Power of a Power Power of a Product Power of a Power
6
Commutative Product of Powers
GuidedPractice 2 1 2 2 3⎣⎡( 5. Simplify _ a b 4b) 2⎤⎦ .
)
(2
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R12.
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1. 15
2. 2  3a
5c 3. _
4. 15g 2
5. _r
6. 7b + 9
8. m 4(m 2)
9 2q 2(9q 4)
d
2
Examples 2–3 Simplify each expression. 7. k(k 3)
Example 4
10. (5u 4v)(7u 4v 3)
2 2 11. ⎡⎣(3 2) ⎤⎦
2 13. (4a 4b 9c)
14. (2f 2g 3h 2)
12. (xy 4) 3
6
15. (3p 5t 6)
16. GEOMETRY The formula for the surface area of a cube is SA = 6s 2, where SA is the surface area and s is the length of any side. a. Express the surface area of the cube as a monomial. b. What is the surface area of the cube if a = 3 and b = 4?
Example 5
Simplify each expression. 2
3
17. (5x 2y) (2xy 3z) (4xyz) 2
19. (2g 3h)(3gj 4) (ghj) 2
404  Lesson 71  Multiplying Monomials
2 18. (3d 2f 3g) ⎡⎣(3d 2f)3⎤⎦2 2 3 20. (7ab 4c)3⎡⎣(2a 2c) ⎤⎦
4
3
ab
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 21. 122
22. 3a 4
23. 2c + 2
2g 24. _ 4h
5k 25. _
26. 6m + 3n
28. (2u 2)(6u 6)
29. (9w 2x 8)(w 6x 4)
31. (b 8c 6d 5)(7b 6c 2d)
32. (14fg 2h 2)(3f 4g 2h 2)
10
Examples 2–3 Simplify each expression. 27 30. 33.
(q 2)(2q 4) (y 6z 9)(6y 4z 2) (j 5k 7) 4
2 2 35. ⎡⎣(2 2) ⎤⎦
3 2 37. ⎡⎣(4r 2t) ⎤⎦
3 2 38. ⎡⎣(2xy 2) ⎤⎦
4
2 4 36. ⎡⎣(3 2) ⎤⎦
Example 4
34. (n 3p)
GEOMETRY Express the area of each triangle as a monomial. 39.
40. 2 5
2g h
2 4
8c d 3
5c d
Example 5
3gh
Simplify each expression. 3 4 41. (2a 3) (a 3)
2 2 42. (c 3) (3c 5)
3 2 43. (2gh 4)3⎡⎣(2g 4h) ⎤⎦
44. (5k 2m)3⎡⎣(4km 4)2⎤⎦
2
4
2
3
45. (p 5r 2) (7p 3r 4) (6pr 3)
46. (5x 2y) (2xy 3z) (4xyz)
47. (5a 2b 3c 4)(6a 3b 4c 2)
48. (10xy 5z 3)(3x 4y 6z 3)
2 49. (0.5x 3)
50. (0.4h 5) 3
3 51. _ c
( 4)
3
4 2 52. _ a
2
(5 ) 4 1 54. (_ m (49m)(17p)(_ p 7 ) 34 ) 2
3 4 53. (8y 3)(3x 2y 2) _ xy
(8 )
B
2
3 4 3 3 55. (3r 3w 4) (2rw)2(3r 2) (4rw 2) (2r 2w 3)
5
3 3 2 2 56. (3ab 2c)2(2a 2b 4) (a 4c 2) (a 2b 4c 5) (2a 3b 2c 4)
57. FINANCIAL LITERACY Cleavon has money in an account that earns 3% simple interest. The formula for computing simple interest is I = Prt, where I is the interest earned, P represents the principal that he put into the account, r is the interest rate (in decimal form), and t represents time in years. a. Cleavon makes a deposit of $2c and leaves it for 2 years. Write a monomial that represents the interest earned. b. If c represents a birthday gift of $250, how much will Cleavon have in this account after 2 years? GEOMETRY Express the volume of each solid as a monomial. 58.
59.
x
2x 3x
2
5x
3
3x
2
60. 4x
2
2x
2
2x
4
3
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61 PACKAGING For a commercial art class, Aiko must design a new container for individually wrapped pieces of candy. The shape that she chose is a cylinder. The formula for the volume of a cylinder is V = πr 2h. a. The radius that Aiko would like to use is 2p 3, and the height is 4p 3. Write a monomial that represents the volume of her container. b. Make a table of values for five possible radius widths and heights if the volume is to remain the same. c. What is the volume of Aiko’s container if the height is doubled? 62. ENERGY Matter can be converted completely into energy by using the formula E = mc 2. Energy E is measured in joules, mass m in kilograms, and the speed c of light is about 300 million meters per second. a. Complete the calculations to convert 3 kilograms of gasoline completely into energy. b. What happens to the energy if the amount of gasoline is doubled? MULTIPLE REPRESENTATIONS In this problem, you will explore exponents.
63.
a. Tabular Copy and use a calculator to complete the table. Power
34
33
32
31
30
Value
3 1
3 2
3 3
3 4
_1
_1
1 _
1 _
3
9
27
81
b. Analytical What do you think the values of 5 0 and 5 1 are? Verify your conjecture using a calculator. c. Analytical Complete: For any nonzero number a and any integer n, a n = ______. d. Verbal Describe the value of a nonzero number raised to the zero power.
H.O.T. Problems C
Use HigherOrder Thinking Skills
64. CHALLENGE For any nonzero real numbers a and b and any integers m and t, m 2t
( b)
a simplify the expression _ t
and describe each step.
65. REASONING Copy the table below. Equation
Related Expression
Power of x
Linear or Nonlinear
y=x y = x2 y = x3
a. For each equation, write the related expression and record the power of x. b. Graph each equation using a graphing calculator. c. Classify each graph as linear or nonlinear. d. Explain how to determine whether an equation, or its related expression, is linear or nonlinear without graphing. 66. OPEN ENDED Write three different expressions that can be simplified to x 6. 67. WRITING IN MATH Write two formulas that have monomial expressions in them. Explain how each is used in a realworld situation.
406  Lesson 71  Multiplying Monomials
SPI 3108.4.10, SPI 3102.5.4, SPI 3102.1.5
Standardized Test Practice 70. CARS In 2002, the average price of a new domestic car was $19,126. In 2008, the average price was $28,715. Based on a linear model, what is the predicted average price for 2014?
68. Which of the following is not a monomial? 1 C _ 2b 3
A 6xy 1 2 B _ a
D 5gh 4
2
69. GEOMETRY The accompanying diagram shows the transformation of XYZ to X’Y’Z’.
A $45,495
C $35,906
B $38,304
D $26,317
9
71. SHORT RESPONSE If a line has a positive slope and a negative yintercept, what happens to the xintercept if the slope and the yintercept are both doubled?
9’ ; ;’
:
y
:’
This transformation is an example of a F G H J
dilation line reflection rotation translation
x
0
Spiral Review Solve each system of inequalities by graphing. (Lesson 68) 72. y < 4x
73. y ≥ 2
2x + 3y ≥ 21
74. y > 2x  1
2y + 2x ≤ 4
75. 3x + 2y < 10
2y ≤ 3x + 2
2x + 12y < 6
Perform the indicated matrix operations. If an operation cannot be performed, write impossible. (Lesson 67)
⎡ 2 76. ⎢5 ⎣ 4
5 1 4
3 ⎤ ⎡ 8 10 + ⎢ 3 0 ⎦ ⎣ 6
2 6 10
6 ⎤ 1 6⎦
⎡11 0 7⎤  [3 0 4] ⎣ 8 11 10 ⎦
77. ⎢
⎡ 2 5 7 ⎤
⎡5 5⎤ 2 11⎤ ⎡2 78. ⎢ +⎢ ⎦ ⎣ 2 2 ⎣ 3 9⎦ 1
79.
⎢ 1
11
⎣ 6 3
⎡ 4
1 + ⎢ 12 4 ⎦ ⎣ 12
0 12 0
9⎤ 8 8⎦
80. BABYSITTING Alexis charges $10 plus $4 per hour to babysit. Alexis needs at least $40 more to buy a television for which she is saving. Write an inequality for this situation. Will she be able to get her television if she babysits for 5 hours? (Lesson 56)
Skills Review Find each quotient. (Lesson 0–3) 81. 64 ÷ (8)
82. 78 ÷ 1.3
83. 42.3 ÷ (6)
84. 23.94 ÷ 10.5
85. 32.5 ÷ (2.5)
86. 98.44 ÷ 4.6 connectED.mcgrawhill.com
407
Dividing Monomials Then
Now
Why?
You multiplied monomials.
1 2
The tallest redwood tree is 112 meters or about 10 2 meters tall. The average height of a redwood tree is 15 meters. The closest power of ten to 15 is 10 1, so an average redwood is about 10 1 meters tall. The ratio of the tallest tree’s height to the average
(Lesson 71)
Find the quotient of two monomials. Simplify expressions containing negative and zero exponents.
2
10 or 10 1. This means the tree’s height is _ 1 10
tallest redwood tree is approximately 10 times as tall as the average redwood tree.
NewVocabulary zero exponent negative exponent order of magnitude
1 Quotients of Monomials
We can use the principles for reducing fractions to 2
10 find quotients of monomials like _ . In the following examples, look for a 10 1 pattern in the exponents. 7 factors 1
1
1
4 factors 1
1
24
Tennessee Curriculum Standards ✔ 3102.3.4 Simplify expressions using exponent rules including negative exponents and zero exponents.
t3
2 ·2 ·2 ·2 1
1
1
1
1
t4 t ·t ·t ·t _ = __ =t
27 2 ·2 ·2 ·2 ·2·2·2 _ = __ = 2 · 2 · 2 or 2 3 1
t ·t ·t 1
4 factors
1
1
3 factors
These examples demonstrate the Quotient of Powers Rule.
KeyConcept Addition Properties Words
To divide two powers with the same base, subtract the exponents.
Symbols
a mp For any nonzero number a, and any integers m and p, _ . p =a
m
c _ = c 11  8 or c 3 11
Examples
c8
_r = r5  2 = r3 5
a
r2
Example 1 Quotient of Powers g h _ . Assume that no denominator equals zero. 3 5
Simplify
gh 2
( )( )
3
3
g h5 g h5 _ = _ _ g
gh 2
h2
= (g 3  1)(h 5  2) =
g 2h 3
Group powers with the same base. Quotient of Powers Simplify.
GuidedPractice Simplify each expression. Assume that no denominator equals zero. 3 4
x y 1A. _ 2 x y
408  Lesson 72
7
10
k m p 1B. _ 3 k 5m p
We can use the Product of Powers Rule to find the powers of quotients for monomials. In the following example, look for a pattern in the exponents. 3 factors
3 factors 3
3
3·3·3 3 =_ (_34 ) = (_34 )(_34 )(_34 ) = _ 4·4·4 4 3
3 factors 2 factors 2 factors 2
c·c c =_ (_dc ) = (_dc )(_dc ) = _ d·d d 2
2
2 factors
StudyTip Power Rules with Variables The power rules apply to variables as well as numbers. For example,
KeyConcept Power of a Quotient To find the power of a quotient, find the power of the numerator and the power of the denominator.
Words
3
(3a) 3a 3 _ 27a 3 _ = or _ . 3 3
( 4b )
(4b)
64b
a For any real numbers a and b ≠ 0, and any integer m, _
Symbols
4 (_3 ) = _3
(_r )5 = _r
4
Examples
5
(b)
5
t
54
m
am = _ m. b
t5
Example 2 Power of a Quotient Simplify
(_) . 3p 3 2 7 3 2
(3p ) 3p _ =_ 2 7
Power of a Quotient
=
Power of a Product
3 2
( )
7 2 3 2 3 (p ) _ 72
9p 6 49
=_
Power of a Power
GuidedPractice
RealWorldCareer
Simplify each expression.
Astronomer An astronomer studies the universe and analyzes space travel and satellite communications. To be a technician or research assistant, a bachelor’s degree is required.
(4)
3x 4 2A. _
3
( ) 5
5x y 2B. _
2
2 2
( )
( )
2y 2C. _3
6
4x 3 2D. _ 4
3z
3
5y
A calculator can be used to explore expressions with 0 as the exponent. There are two methods to explain why a calculator gives a value of 1 for 3 0. Method 1 5
3 _ = 35  5 3
5
= 30
Method 2 Quotient of Powers Simplify.
35 3 ·3 ·3 ·3 ·3 _ = __ 35
3 ·3 ·3 ·3 ·3
=1
Definition of powers Simplify.
5
3 can only have one value, we can conclude that 3 0 = 1. A zero exponent is Since _ 5 3
any nonzero number raised to the zero power. connectED.mcgrawhill.com
409
KeyConcept Zero Exponent Property Words
Any nonzero number raised to the zero power is equal to 1.
Symbols
For any nonzero number a, a 0 = 1.
Examples
15 0 = 1
(_bc )0 = 1
(_27 )
0
=1
Example 3 Zero Exponent Simplify each expression. Assume that no denominator equals zero. 4n 2q 5r 2 0
( ) ( _)
a. _ 3 2 9n q r

4n 2q 5r 2 0 9n 3q 2r
=1
a0 = 1
5 0
x y b. _ 3 x
5 0
5
x3
x3
x y x (1) _ =_
a0 = 1
= x2
Quotient of Powers
StudyTip
GuidedPractice
Zero Exponent Be careful of parentheses. The expression (5x) 0 is 1 but 5x 0 = 5.
b 4c 2d 0 3A. _ 2
(
4 7 3
2f g h 3B. _ 9
b c
15f 3g h 6
)
0
2 Negative Exponents_
Any nonzero real number raised to a negative power is a negative exponent. To investigate the meaning of a negative exponent, we can
2 simplify expressions like c 5 using two methods.
c
Method 1 c2 _ = c2  5 c
5
= c 3
Method 2 c2 c ·c _ = __
Quotient of Powers
c5
Simplify.
Definition of powers
c ·c ·c·c·c _ = 13 c
Simplify.
2
c 1 Since _ can only have one value, we can conclude that c 3 = _ . 5 3 c
c
KeyConcept Negative Exponent Property Words
For any nonzero number a and any integer n, an is the reciprocal of an. Also, the reciprocal of an is an.
Symbols
For any nonzero number a and any integer n, a n = 1n
Examples
410  Lesson 72  Dividing Monomials
_
2
4
1 1 =_ =_ 24
16
1 _ = j4 j 4
a
An expression is considered simplified when it contains only positive exponents, each base appears exactly once, there are no powers of powers, and all fractions are in simplest form.
Example 4 Negative Exponents Simplify each expression. Assume that no denominator equals zero. n p _ 5 4
a.
r 2 n 5p 4 _ r 2
( 1 )( 1 )( r ) p _ 1 _ = (_ )r ( ) 1 (1) n p4 _ n 5 _ 1 = _ 2 4
Write as a product of fractions.
2
p 4r 2
=_ 5
Negative Signs Be aware of where a negative sign is placed.
_
_
5
5
b.
_
a
a
Multiply.
n
StudyTip
_
a n = 1n and 1n = a n
5
5r t _ 3 4
20r 2t 7u 5
( )( )( )( )
5 5r 3t 4 r 3 _ t4 _ 1 _ _ = _
5 1 = 1 , while 5 1 ≠ 1 .
20r 2t 7u 5
20
r2
t7
Group powers with the same base.
u 5
1 ( 3  2)( 4  7)( 5) = _ r t u
Quotient of Powers and Negative Exponents Property
1 5 3 5 = _ r t u
Simplify.
1 _ 1 ( 5) 1 _ = _ u 5 3
Negative Exponent Property
( 4) 4
( )( t )
4 r 5
u = _ 5 3
Multiply.
4r t
c.
2a b c __ 2 3 5
10a 3b 1c 4
( 10 )( a )( b )( c ) 1 ( )(b = (_ a 5)
a2 _ b3 _ c 5 2 _ 2a 2b 3c 5 __ = _ 10a 3b 1c 4
3
1
2 ( 3)
Group powers with the same base.
4
3 ( 1))( 5  ( 4))
c
1 5 4 1 =_ a b c
Simplify.
5
1 ( 5)( 4) _ =_ a b 1c 5
Quotient of Powers and Negative Exponents Property
()
Negative Exponent Property
5 4
a b =_
Multiply.
5c
GuidedPractice Simplify each expression. Assume that no denominator equals zero.
RealWorldLink An adult human weighs about 70 kilograms and an adult dairy cow weighs about 700 kilograms. Their weights differ by 1 order of magnitude.
v 3wx 2 4A. _ 6 wy
32a 8b 3c 4 4B. _ 3 2 4a b 5c
3 2
4C.
6
5j k m _ 25k 4m 2
O Order of magnitude is used to compare measures and to estimate and perform rough ccalculations. The order of magnitude of a quantity is the number rounded to the n nearest power of 10. For example, the power of 10 closest to 95,000,000,000 is 10 11, o or 100,000,000,000. So the order of magnitude of 95,000,000,000 is 10 11. connectED.mcgrawhill.com
411
RealWorld Example 5 Apply Properties of Exponents HEIGHT Suppose the average height of a man is about 1.7 meters, and the average height of an ant is 0.0008 meter. How many orders of magnitude as tall as an ant is a man? Understand We must find the order of magnitude of the heights of the man and ant. Then find the ratio of the orders of magnitude of the man’s height to that of the ant’s height.
RealWorldLink
Plan Round each height to the nearest power of ten. Then find the ratio of the height of the man to the height of the ant.
There are over 14,000 species of ants living all over the world. Some ants can carry objects that are 50 times their own weight.
Solve The average height of a man is close to 1 meter. So, the order of magnitude is 10 0 meter. The average height of an ant is about 0.001 meter. So, the order of magnitude is 10 3 meters.
Source: Maine Animal Coalition
0
10 The ratio of the height of a man to the height of an ant is about _ . 3 10 0 _ = 10 0  (3) 10 3
10
Quotient of Powers
= 10 3
0  (3) = 0 + 3 or 3
= 1000
Simplify.
So, a man is approximately 1000 times as tall as an ant, or a man is 3 orders of magnitude as tall as an ant. Check The ratio of the man’s height to the ant’s height is 1.7 _ = 2125. The order of magnitude of 2125 is 10 3. 0.0008
GuidedPractice 5. ASTRONOMY The order of magnitude of the mass of Earth is about 10 27. The order of magnitude of the Milky Way galaxy is about 10 44. How many orders of magnitude as big is the Milky Way galaxy as Earth?
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–4 Simplify each expression. Assume that no denominator equals zero. t 5u 4 1. _ 2 t u
g h m 5. _ 7
(
)
3
2
(
12. _ 3 4 x yz
)
3 2
f g 15. _ 4 h
Example 5
(
4
( )
5
5g
4
)
3f gh 13. _ 4 32f 3g h
0
2 8 5
8x y z 16. _ 4 7 7 12x y
q n w
2c 3d 5 11. _ 2
t
6
n q w 8. _ 2 3
z x y
r 3v 2 10. _ 7
3xy 4z 2 0
4 4
x y z 7. _ 5 2
t v
2a 3b 5 9. _
b c f
3 2 6
r 4t 7v 2 6. _ 7 2
hg
b c f 4. _ 4 3 5
m r p
a b c
8 2
4 6 8
6 5 3
m r p 3 _ 5 2 3
a 6b 4c 10 2. _ 3 2
z
4r 2v 0t 5 14. _ 3 2rt
2a 2b 7c 10 17. _ 3 2 3 6a
b c
18. FINANCIAL LITERACY The gross domestic product (GDP) for the United States in 2008 was $14.204 trillion, and the GDP per person was $47,580. Use order of magnitude to approximate the population of the United States in 2008.
412  Lesson 72  Dividing Monomials
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–4 Simplify each expression. Assume that no denominator equals zero. 4 2
12 3
m p 19. _ 2
p t r 20. _ 2
m p
p tr
4 2
( )
4 4 3
c d f 22. _ 2 c
3xy 23. _ 2
d 4f 3
5z
2 7 3
( )
4 9
p t 25. _
x y 26. _ 2
10
3 2
( ) ( _)
3np 28. _ 2
29
7q
31. 
5f 9g 4h 2 0
4
3 2
)
Example 5
B
b
c
2
( )
4
4p
4 3
r
4 2
12m p 36. _ 3 9 15m p
39t 4uv 2 39. _ 3 7
r 3t 1x 5 41. _ 5
g h j 42. _ 5 0 2
tx
0
a bc
14f g h 38. _ 3 21k
a 2b 4c 5 40. _ 4 4 3 a
p t 33. _ 2
3 2 7
k 2m
(
12 7 2
8f 2g
k m p 37. _ 2
9tuv
3m 5r 3 30. _ 8
3 2 0
8cd f
)
2r 3t 6 4 5u 9
(_)
2f g h 35. _ 2
2 5
(
3t 6u 2v 5 24. _ 21
z
p t r
5c d 34. _ 5 0
12t
a 7b 8c 8 27. _ 5 7
p t r 32. _ 2 7
fg 2h 3
3 4 2
3m r p 21. _ 4
13t
u
0 7 2
g
h j
43. INTERNET In a recent year, there were approximately 3.95 million Internet hosts. Suppose there were 208 million Internet users. Determine the order of magnitude for the Internet hosts and Internet users. Using the orders of magnitude, how many Internet users were there compared to Internet hosts? 1 44. PROBABILITY The probability of rolling a die and getting an even number is _ . 2 If you roll the die twice, the probability of getting an even number both times 1 _ 1 1 2 is _ or _ . Write an expression to represent the probability of rolling a
( 2 )( 2 ) ( 2 )
die d times and getting an even number every time. Write the expression as a power of 2. Simplify each expression. Assume that no denominator equals zero. 13r 7 46. _ 4
4w 12 45. _ 3 12w
39r
2
2 5
3wy 48. _3
20qr t 49. _ 0 4 2
(w 1y) (2g 3h 2) 2 _ 51. (g 2h 0) 3
(
4q r t
2 2
(5pr ) 52. _ 3
)
2a 2b 4c 2 54. __ 2 5 7 4a
b
c
1
(3p 1r) (16x 2y 1) 0 55. _ (4x 0y 4z) 2
(4k 3m 2) 3 47. _ (5k 2m 3) 2 3 0 2
12c d f 50. _ 5 3 4 6c d
f
6 1 2 2
(
)
3x y z 53. __ 2 5
(
6x
yz
0 2 3
)
4 c d f 56. _ 4 5 2c
d
3
57. COMPUTERS The processing speed of an older desktop computer is about 10 8 instructions per second. A new computer can process about 10 10 instructions per second. The newer computer is how many times as fast as the older one? connectED.mcgrawhill.com
413
58. ASTRONOMY The brightness of a star is measured in magnitudes. The lower the magnitude, the brighter the star. A magnitude 9 star is 2.51 times as bright as a magnitude 10 star. A magnitude 8 star is 2.51 · 2.51 or 2.51 2 times as bright as a magnitude 10 star. a. How many times as bright is a magnitude 3 star as a magnitude 10 star? b. Write an expression to compare a magnitude m star to a magnitude 10 star. c. Magnitudes can be measured in negative numbers. Does your expression hold true? Give an example or counterexample. 1 59 PROBABILITY The probability of rolling a die and getting a 3 is _. If you roll the die 1 _ 1 2 twice, the probability of getting a 3 both times is _ · 1 or _ . 6
6
(6)
6
a. Write an expression to represent the probability of rolling a die d times and getting a 3 each time. b. Write the expression as a power of 6.
C
60.
MULTIPLE REPRESENTATIONS To find the area of a circle, use A = πr 2. The formula for the area of a square is A = s 2. a. Algebraic Find the ratio of the area of the circle to the area of the square. b. Algebraic If the radius of the circle and the length of each side of the square are doubled, find the ratio of the area of the circle to the square. c. Tabular Copy and complete the table. Radius
Area of Circle
Area of Square
Ratio
r 2r 3r 4r 5r 6r
d. Analytical What conclusion can be drawn from this?
H.O.T. Problems
Use HigherOrder Thinking Skills
61. REASONING Is x y · x z = x yz sometimes, always, or never true? Explain. 62. OPEN ENDED Name two monomials with a quotient of 24a 2b 3. 1 63. CHALLENGE Use the Quotient of Powers Property to explain why x n = _ n. x
0
64. REASONING Write a convincing argument to show why 3 = 1 using the following pattern: 3 5 = 243, 3 4 = 81, 3 3 = 27, 3 2 = 9. 65. WRITING IN MATH Explain how to use the Quotient of Powers property and the Power of a Quotient property.
414  Lesson 72  Dividing Monomials
r
SPI 3108.4.7, SPI 3102.3.6, SPI 3102.5.5, SPI 3102.1.3
Standardized Test Practice 66. Geometry What is the perimeter of the figure in meters? A B C D
40x 80x 160x 400x
68. EXTENDED RESPONSE Jessie and Jonas are playing a game using the spinners below. Each spinner is equally likely to stop on any of the four numbers. In the game, a player spins both spinners and calculates the product of the two numbers on which the spinners have stopped.
8x
12x
20x
67. In researching her science project, Leigh learned that light travels at a constant rate and that it takes 500 seconds for light to travel the 93 million miles from the Sun to Earth. Mars is 142 million miles from the Sun. About how many seconds will it take for light to travel from the Sun to Mars? F G H J
4
1
4
1
3
2
3
2
a. What product has the greatest probability of occurring? b. What is the probability of that product occurring? 3
69. Simplify (4 2 · 5 0 · 64) .
235 seconds 327 seconds 642 seconds 763 seconds
1 A _
C 320
B 64
D 1024
64
Spiral Review 70. GEOLOGY The seismic waves of a magnitude 6 earthquake are 10 2 times as great as a magnitude 4 earthquake. The seismic waves of a magnitude 4 earthquake are 10 times as great as a magnitude 3 earthquake. How many times as great are the seismic waves of a magnitude 6 earthquake as those of a magnitude 3 earthquake? (Lesson 71) Solve each system of inequalities by graphing. (Lesson 68) 71. y ≥ 1 x < 1
72. y ≥ 3 yx<1
73. y < 3x + 2 y ≥ 2x + 4
74. y  2x < 2 y  2x > 4
Solve each inequality. Check your solution. (Lesson 53) 75. 5(2h  6) > 4h
76. 22 ≥ 4(b  8) + 10
77. 5(u  8) ≤ 3(u + 10)
78. 8 + t ≤ 3(t + 4) + 2
79. 9n + 3(1  6n) ≤ 21
80. 6(b + 5) > 3(b  5)
81. GRADES In a high school science class, a test is worth three times as much as a quiz. What is the student’s average grade? (Lesson 29)
Science Grades Tests
Quizzes
85 92
82 75 95
Skills Review Evaluate each expression. (Lesson 11) 82. 9 2
83. 11 2
84. 10 6
85. 10 4
86. 3 5
87. 5 3
88. 12 3
89. 4 6 connectED.mcgrawhill.com
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Scientific Notation Then
Now
Why?
You found products and quotients of monomials.
1 2
Space tourism is a multibillion dollar industry. For a price of $20 million, a civilian can travel on a rocket or shuttle and visit the International Space Station (ISS) for a week.
(Lessons 71 and 72)
NewVocabulary scientific notation
Tennessee Curriculum Standards ✔ 3102.2.5 Perform operations with numbers in scientific notation. ✔ 3102.2.6 Use appropriate technologies to apply scientific notation to realworld problems. SPI 3102.2.2 Multiply, divide, and square numbers expressed in scientific notation.
Express numbers in scientific notation. Find products and quotients of numbers expressed in scientific notation.
1 Scientific Notation
Very large and very small numbers such as $20 million can be cumbersome to use in calculations. For this reason, numbers are often expressed in scientific notation. A number written in scientific notation is of the form a × 10 n, where 1 ≤ a < 10 and n is an integer.
KeyConcept Order of Operations Step 1 Move the decimal point until it is to the right of the first nonzero digit. The result is a real number a. Step 2 Note the number of places n and the direction that you moved the decimal point. Step 3 If the decimal point is moved left, write the number as a × 10 n. If the decimal point is moved right, write the number as a × 10 n. Step 4 Remove the unnecessary zeros.
Example 1 Standard Form to Scientific Notation Express each number in scientific notation. a. 201,000,000 Step 1 201,000,000 2.01000000 a = 2.01000000 Step 2 The decimal point moved 8 places to the left, so n = 8. Step 3 201,000,000 = 2.01000000 × 10 8 Step 4 2.01 × 10 8 b. 0.000051 Step 1 0.000051
00005.1
a = 00005.1
Step 2 The decimal point moved 5 places to the right, so n = 5. Step 3 0.000051 = 00005.1 × 10 5 Step 4 5.1 × 10 5
GuidedPractice 1A. 68,700,000,000
416  Lesson 73
1B. 0.0000725
You can also rewrite numbers in scientific notation in standard form.
WatchOut! Negative Signs Be careful about the placement of negative signs. A negative sign in the exponent means that the number is between 0 and 1. A negative sign before the number means that it is less than 0.
KeyConcept Scientific Notation to Standard Form Step 1 In a × 10 n, note whether n > 0 or n < 0. Step 2 If n > 0, move the decimal point n places right. If n < 0, move the decimal point n places left. Step 3 Insert zeros, decimal point, and commas as needed for place value.
Example 2 Scientific Notation to Standard Form Express each number in standard form. a. 6.32 × 10 9 Step 1 The exponent is 9, so n = 9. Step 2 Since n > 0, move the decimal point 9 places to the right. 6320000000 6.32 × 10 9 Step 3 6.32 × 10 9 = 6,320,000,000
Rewrite; insert commas.
b. 4 × 10 7 Step 1 The exponent is 7, so n = 7. Step 2 Since n < 0, move the decimal point 7 places to the left. 0000004 4 × 10 7 Step 3 4 × 10 7 = 0.0000004
Rewrite; insert a 0 before the decimal point.
GuidedPractice Simplify each expression. 2B. 9.03 × 10 5
2A. 3.201 × 10 6
ProblemSolvingTip
Example 3 Multiply with Scientific Notation
Estimate Reasonable Answers Estimating an answer before computing the solution can help you determine if your answer is reasonable.
Evaluate (3.5 × 10 3)(7 × 10 5). Express the result in both scientific notation and standard form.
(3.5 × 10 3)(7 × 10 5) = (3.5 × 7)(10 3 × 10 5)
Original expression
= 24.5 × 10 2 = (2.45 × 10 1) × 10 2 = 2.45 × 10 3 = 2450
Product of Powers
Commutative and Associative Properties 24.5 = 2.45 × 10 Product of Powers Standard form
GuidedPractice Evaluate each product. Express the results in both scientific notation and standard form. 3A. (6.5 × 10 12)(8.7 × 10 15)
3B. (1.95 × 10 8)(7.8 × 10 2) connectED.mcgrawhill.com
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Example 4 Divide with Scientific Notation Evaluate
StudyTip Quotient of Powers Recall that the Quotient of Powers Property is only valid for powers that have the same base. Since 10 8 and 10 3 have the same base, the property applies.
3.066 × 10 _ . Express the result in both scientific notation and 8
7.3 × 10 3
standard form.
( 7.3 ) ( 10 )
3.066 × 10 8 3.066 _ 10 8 _ = _ 7.3 × 10 3
Product rule for fractions
3
= 0.42 × 10 5 = 4.2 × 10
1
Quotient of Powers
× 10
5
0.42 = 4.2 × 10 1
= 4.2 × 10 4
Product of Powers
= 42,000
Standard form
GuidedPractice Evaluate each quotient. Express the results in both scientific notation and standard form. 2.3958 × 10 3 4A. __ 8
1.305 × 10 3 4B. _ 4
1.98 × 10
1.45 × 10
RealWorld Example 5 Use Scientific Notation MUSIC In the United States, a CD reaches gold status once 500 thousand copies are sold. A CD reaches platinum status once 1 million or more copies are sold. a. Express the number of copies of CDs that need to be sold to reach each status in standard notation. gold status: 500 thousand = 500,000; platinum status: 1 million = 1,000,000
RealWorldLink The platinum award was created in 1976. In 2004, the criteria for the award was extended to digital sales. The topselling artist of all time is the Beatles with 170 million units sold. Source: Recording Industry Association of America
b. Write each number in scientific notation. gold status: 500,000 = 5 × 10 5; platinum status: 1,000,000 = 1 × 10 6 c. How many copies of a CD have sold if it has gone platinum 13 times? Write your answer in scientific notation and standard form. A CD reaches platinum status once it sells 1 million records. Since the CD has gone platinum 13 times, we need to multiply by 13. (13)(1 × 10 6)
Original expression
= (13 × 1)(10 = 13 × 10
6)
6
= (1.3 × 10 1) × 10 6 = 1.3 × 10
7
= 13,000,000
Associative Property 13 × 1 = 13 13 = 1.3 × 10 Product of Powers Standard form
GuidedPractice 5. SATELLITE RADIO Suppose a satellite radio company earned $125.4 million in one year. A. Write this number in standard form. B. Write this number in scientific notation. C. If the following year the company earned 2.5 times the amount earned the previous year, determine the amount earned. Write your answer in scientific notation and standard form.
418  Lesson 73  Scientific Notation
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R12.
Express each number in scientific notation. 1. 185,000,000
2. 1,902,500,000
3. 0.000564
4. 0.00000804
MONEY Express each number in scientific notation. 5. Teenagers spend $13 billion annually on clothing. 6. Teenagers have an influence on their families’ spending habit. They control about $1.5 billion of discretionary income. Example 2
Express each number in standard form. 7. 1.98 × 10 7
8. 4.052 × 10 6
9. 3.405 × 10 8 Example 3
10. 6.8 × 10 5
Evaluate each product. Express the results in both scientific notation and standard form. 12. (7.08 × 10 14)(5 × 10 9) 14. (2.9 × 10 2)(5.2 × 10 9)
11. (1.2 × 10 3)(1.45 × 10 12) 13. (5.18 × 10 2)(9.1 × 10 5) Example 4
Evaluate each quotient. Express the results in both scientific notation and standard form. 1.035 × 10 8 15. _ 4
2.542 × 10 5 16. _ 10
2.3 × 10
4.1 × 10
7
2.05 × 10 8 18. _ 2
1.445 × 10 17. __ 5 1.7 × 10
Example 5
4 × 10
19. AIR FILTERS Salvador bought an air purifier to help him deal with his allergies. The filter in the purifier will stop particles as small as one hundredth of a micron. A micron is one millionth of a millimeter. a. Write one hundredth and one micron in standard form. b. Write one hundredth and one micron in scientific notation. c. What is the smallest size particle in meters that the filter will stop? Write the result in both standard form and scientific notation.
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Express each number in scientific notation. 20. 1,220,000
21 58,600,000
22. 1,405,000,000,000
23. 0.0000013
24. 0.000056
25. 0.000000000709
EMAIL Express each number in scientific notation. 26. Approximately 100 million emails sent to the President are put into the National Archives. 27. By 2015, the email security market will generate $6.5 billion. Example 2
Express each number in standard form. 28. 1 × 10 12
29. 9.4 × 10 7
30. 8.1 × 10 3
31. 5 × 10 4
32. 8.73 × 10 11
33. 6.22 × 10 6 connectED.mcgrawhill.com
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Example 2
INTERNET Express each number in standard form. 34. About 2.1 × 10 7 people, aged 12 to 17, use the Internet. 35. Approximately 1.1 × 10 7 teens go online daily.
Examples 3–4 Evaluate each product or quotient. Express the results in both scientific notation and standard form. 36. (3.807 × 10 3)(5 × 10 2)
9.6 × 10 3 37. _ 4
2.88 × 10 3 38. _ 5
39 (6.5 × 10 7)(7.2 × 10 2)
40. (9.5 × 10 18)(9 × 10 9)
8.8 × 10 3 41. _ 4
9.15 × 10 3 42. _ 6.1 × 10
43. (2.01 × 10 4)(8.9 × 10 3)
44. (2.58 × 10 2)(3.6 × 10 6)
5.6498 × 10 10 45. __ 4
1.363 × 10 16 46. _ 6
47. (9.04 × 10 6)(5.2 × 10 4)
48. (1.6 × 10 5)(2.3 × 10 3)
6.25 × 10 4 49. _ 2
3.75 × 10 9 50. _ 4
51. (3.4 × 10 4)(7.2 × 10 15)
8.6 × 10 4 52. _ 6
53. (6.3 × 10 2)(3.5 × 10 4)
1.2 × 10
2.9 × 10
1.5 × 10
2 × 10
1.2 × 10
4 × 10
8.2 × 10
1.25 × 10
54. ASTRONOMY The distance between Earth and the Sun varies throughout the year. Earth is closest to the Sun in January when the distance is 91.4 million miles. In July, the distance is greatest at 94.4 million miles. a. Write 91.4 million in both standard form and in scientific notation. b. Write 94.4 million in both standard form and in scientific notation. c. What is the percent increase in distance from January to July? Round to the nearest tenth of a percent.
B
Evaluate each product or quotient. Express the results in both scientific notation and standard form. 55. (4.65 × 10 2)(5 × 10 6)
2.548 × 10 5 56. _ 2
2.135 × 10 5 57. _ 12
58. (4.8 × 10 5)(3.16 × 10 5)
59. (4.3 × 10 3)(4.5 × 10 4)
5.184 × 10 5 60. __ 3
3.5 × 10
61. (5 × 10
3)(
1.8 × 10
7)
2.8 × 10
7.2 × 10
1.032 × 10 4 62. __ 5 8.6 × 10
LIGHT The speed of light is approximately 3 × 10 8 meters per second. 63. Write an expression to represent the speed of light in kilometers per second. 64. Write an expression to represent the speed of light in kilometers per hour. 65. Make a table to show how many kilometers light travels in a day, a week, a 30day month, and a 365day year. Express your results in scientific notation. 66. The distance from Earth to the Moon is approximately 3.844 × 10 5 kilometers. How long would it take light to travel from Earth to the Moon?
420  Lesson 73  Scientific Notation
67 EARTH The population of Earth is about 6.623 × 10 9. The land surface of Earth is 1.483 × 10 8 square kilometers. What is the population density for the land surface area of Earth? 68. RIVERS A drainage basin separated from adjacent basins by a ridge, hill, or mountain is known as a watershed. The watershed of the Amazon River is 2,300,000 square miles. The watershed of the Mississippi River is 1,200,000 square miles. a. Write each of these numbers in scientific notation. b. How many times as large is the Amazon River watershed as the Mississippi River watershed? 69. AGRICULTURE In a recent year, farmers planted approximately 92.9 million acres of corn. They also planted 64.1 million acres of soybeans and 11.1 million acres of cotton. a. Write each of these numbers in scientific notation and in standard form. b. How many times as much corn was planted as soybeans? Write your results in standard form and in scientific notation. Round your answer to four decimal places. c. How many times as much corn was planted as cotton? Write your results in standard form and in scientific notation. Round your answer to four decimal places.
H.O.T. Problems B
Use HigherOrder Thinking Skills
70. REASONING Which is greater, 100 10 or 10 100? Explain your reasoning. 71. ERROR ANALYSIS Syreeta and Pete are solving a division problem with scientific notation. Is either of them correct? Explain your reasoning.
Pete
Syreeta
3.65 × 10 = 0.73 × 10 17 _ 12
3.65 × 10 12 _ = 0.73 × 10 17 5 × 10 5
5 × 10 5
= 7.3 × 10 18
= 7.3 × 10 16
72. CHALLENGE Order these numbers from least to greatest without converting them to standard form. 5.46 × 10 3, 6.54 × 10 3, 4.56 × 10 4, 5.64 × 10 4, 4.65 × 10 5 73. REASONING Determine whether the statement is always, sometimes, or never true. Give examples or a counterexample to verify your reasoning. When multiplying two numbers written in scientific notation, the resulting number can have no more than two digits to the left of the decimal point. 74. OPEN ENDED Write two numbers in scientific notation with a product of 1.3 × 10 3. Then name two numbers in scientific notation with a quotient of 1.3 × 10 3. 75. WRITING IN MATH Write the steps that you would use to divide two numbers written in scientific notation. Then describe how you would write the results a for a = 2 × 10 3 and b = 4 × 10 5. in standard form. Demonstrate by finding _ b
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SPI 3102.3.6, SPI 3102.5.1, SPI 3102.1.1
Standardized Test Practice 76. Which number represents 0.05604 × 10 8 written in standard form? A 0.0000000005604 B 560,400
C 5,604,000 D 50,604,000
Distance from School
77. Toni left school and rode her bike home. The graph below shows the relationship between her distance from the school and time. 1 0.75 0.5 0.25 0
y
(30, 1)
(40, 1)
(0, 0)
5 10 15 20 25 30 35 40 45 50 x
Time (minutes)
78. SHORT RESPONSE In his first four years of coaching football, Coach Delgato’s team won 5 games the first year, 10 games the second year, 8 games the third year, and 7 games the fourth year. How many games does the team need to win during the fifth year to have an average of 8 wins per year? 79. The table shows the relationship between Calories and grams of fat contained in an order of fried chicken from various restaurants. Calories
305
410
320
500
510
440
Fat (g)
28
34
28
41
42
38
Assuming that the data can best be described by a linear model, about how many grams of fat would you expect to be in a 275Calorie order of fried chicken?
Which explanation could account for the section of the graph from x = 30 to x = 40? F Toni rode her bike down a hill. G Toni ran all the way home. H Toni stopped at a friend’s house on her way home. J Toni returned to school to get her mathematics book.
A B C D
22 25 27 28
Spiral Review Simplify. Assume that no denominator is equal to zero. (Lesson 72) 89 80. _ 6
65 81. _ 3
8
r 8t 12 82. _ 2 7 r t
6
5d 3g 2 2
2 4 3
( )
3a 4b 4 4
( )
83. _ 2
( )
4n p 85. _ 3
84. _ 4
8c
8p
3h
86. CHEMISTRY Lemon juice is 10 2 times as acidic as tomato juice. Tomato juice is 10 3 times as acidic as egg whites. How many times as acidic is lemon juice as egg whites? (Lesson 71) Write each equation in slopeintercept form. (Lesson 42) 87. y  2 = 3(x  1)
88. y  5 = 6(x + 1)
89. y + 2 = 2(x + 5)
1 90. y + 3 = _ (x + 4)
2 91. y  1 = _ (x + 9)
1 92. y + 3 = _ (x + 2)
2
3
4
Skills Review Simplify each expression. If not possible, write simplified. (Lesson 14) 93. 3u + 10u
94. 5a  2 + 6a
95. 6m 2  8m
96. 4w 2 + w + 15w 2
97. 13(5 + 4a)
98. (4t  6)16
422  Lesson 73  Scientific Notation
Algebra Lab
Polynomials Algebra tiles can be used to model polynomials. A polynomial is a monomial or the sum of monomials. The diagram below shows the models.
Tennessee Curriculum Standards ✔ 3102.1.11 Use manipulatives to model algebraic concepts.
Polynomial Models • Polynomials are modeled using three types of tiles.
• Each tile has an opposite.
2
1
Y
Y
1
Y
Y
2
Activity Use algebra tiles to model each polynomial. • 5x Y
To model this polynomial, you will need 5 green xtiles.
Y
Y
Y
Y
• 3x 2  1 Y
To model this polynomial, you will need 3 blue x 2tiles and 1 red 1tile.
2
Y
2
Y
2
1
• 2x 2 + x + 3 2
Y
To model this polynomial, you will need 2 red x 2tiles, 1 green xtile, and 3 yellow 1tiles.
Y
2
Y 1
1
1
Model and Analyze Use algebra tiles to model each polynomial. Then draw a diagram of your model. 1. 4x 2
2. 3x  5
3. 2x 2  3x
4. x 2 + 2x + 1
Write an algebraic expression for each model. 5.
6. Y
2
Y
2
Y
Y Y Y Y Y
2
Y
2
Y
2
Y
Y 1
7.
8. Y
2
Y
Y
Y
Y
Y 1 1
2
Y 1
1
1
1
1
1
9. MAKE A CONJECTURE Write a sentence or two explaining why algebra tiles are sometimes called area tiles. connectED.mcgrawhill.com
423
Polynomials Then
Now
Why?
You identified monomials and their characteristics.
1 2
In 2017, sales of digital audio players are expected to reach record numbers. The sales data can be modeled by the equation U = 2.7t 2 + 49.4t + 128.7, where U is the number of units shipped in millions and t is the number of years since 2005.
(Lesson 71)
Find the degree of a polynomial. Write polynomials in standard form.
The expression 2.7t 2 + 49.4t + 128.7 is an example of a polynomial. Polynomials can be used to model situations.
NewVocabulary polynomial binomial trinomial degree of a monomial degree of a polynomial standard form of a polynomial leading coefficient
1 Degree of a Polynomial
A polynomial is a monomial or the sum of monomials, each called a term of the polynomial. Some polynomials have special names. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials.
Example 1 Identify Polynomials Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. Expression a. 4y  5xz
Yes; 4y  5xz is the sum of the two monomials 4y and 5xz.
binomial
b. 6.5
Yes; 6.5 is a real number.
monomial
c. 7a 3 + 9b
7 No; 7a 3 = _ , which is not a monomial. 3
none of these
d. 6x 3 + 4x + x + 3
Yes; 6x 3 + 4x + x + 3 = 6x 3 + 5x + 3, the sum of three monomials.
Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. SPI 3102.3.2 Operate with polynomials and simplify results.
Monomial, binomial, or trinomial?
Is it a polynomial?
a
trinomial
GuidedPractice 1A. x
1B. 3y 2  2y + 4y  1
1C. 5rx + 7tuv
1D. 10x 4  8x a
The degree of a monomial is the sum of the exponents of all its variables. A nonzero constant has degree 0. Zero has no degree. The degree of a polynomial is the greatest degree of any term in the polynomial. To find the degree of a polynomial, you must find the degree of each term. Some polynomials have special names based on their degree.
424  Lesson 74
Degree
Name
0
constant
1
linear
2
quadratic
3
cubic
4
quartic
5
quintic
6 or more
6th degree, 7th degree, and so on
ReadingMath Prefixes The prefixes mono, bi, and tri mean one, two, and three, respectively. Hence, a monomial has one term, a binomial has two terms, and a trinomial has three terms.
Example 2 Degree of a Polynomial Find the degree of each polynomial. a. 3a 2b 3 + 6 Step 1 Find the degree of each term. 3a 2b 3: degree = 2 + 3 or 5
6: degree 0
Step 2 The degree of the polynomial is the greatest degree, 5. b. 2d 3  5c 5d  7 2d 3: degree = 3
5c 5d: degree = 5 + 1 or 6
7: degree 0
The degree of the polynomial is 6.
GuidedPractice 2B. 2rt  3rt 2  7r 2t 2  13
2A. 7xy 5z
2 Polynomials in Standard Form
The terms of a polynomial may be written in any order. Polynomials written in only one variable are usually written in standard form. The standard form of a polynomial is written with the terms in order from greatest degree to least degree. When a polynomial is written in standard form, the coefficient of the first term is called the leading coefficient. leading coefficient
greatest degree
4x 3  5x 2 + 2x + 7
Standard form:
Example 3 Standard Form of a Polynomial Write each polynomial in standard form. Identify the leading coefficient. a. 3x 2 + 4x 5  7x Step 1 Find the degree of each term. Degree: Polynomial:
2
5
1
3x 2 + 4x 5  7x
Step 2 Write the terms in descending order: 4x 5 + 3x 2  7x. The leading coefficient is 4. b. 5y  9  2y 4  6y 3 Step 1 Degree: Polynomial:
1
0
4
3
5y  9  2y 4  6y 3
Step 2 2y 4  6y 3 + 5y  9
The leading coefficient is 2.
GuidedPractice 3A. 8  2x 2 + 4x 4  3x
3B. y + 5y 3  2y 2  7y 6 + 10 connectED.mcgrawhill.com
425
A function represented by a polynomial equation is a polynomial function. Polynomial functions can be used to predict values of events before they occur.
RealWorld Example 4 Use a Polynomial BUSINESS From 2003 through 2009, the number U of skateboards (in thousands) produced at a manufacturing plant can be modeled by the function U(t) = 3t 2  2t + 10, where t is the number of years since 2003. How many skateboards were produced in 2005? Find the value of t, and substitute the value of t to find the number of skateboards produced. Since t is the number of years since 2003, t equals 2005  2003 or 2. U(t) = 3t 2  2t + 10 = 3(2) 2  2(2) + 10 = 3(4)  4 + 10 = 12  4 + 10 = 18
RealWorldLink Louisville Extreme Park in Louisville, Kentucky, encompasses 40,000 square feet of skating surface and includes a 24foot fullpipe.
Original equation t=2 Simplify. Multiply. Simplify.
Since U is in thousands, the number of skateboards produced was 18 thousand or 18,000.
Source: Louisville Metro Government
GuidedPractice 4A. How many skateboards were produced in 2008? 4B. If this trend continues, how many skateboards will be produced in 2018?
Check Your Understanding Example 1
Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. 1. 7ab + 6b 2  2a 3
2. 2y  5 + 3y 2
3. 3x 2
4m 4. _
2 3
5. 5m p + 6 Example 2
9. 7z
Example 4
3p
6. 5q 4 + 6q
Find the degree of each polynomial. 7. 3
Example 3
= StepbyStep Solutions begin on page R12.
8. 6p 3  p 4 3 10. _ 4
11 12  7q 2t + 8r
12. 2a 2b 5 + 5  ab
13. 6df 3 + 3d 2f 2 + 2d + 1
14. 9hjk  4h 2j 3 + 5j 2k 2  h 3k 3
Write each polynomial in standard form. Identify the leading coefficient. 15. 2x 5  12 + 3x
16. y 3 + 3y  3y 2 + 2
17. 4z  2z 2  5z 4
18. 2a + 4a 3  5a 2  1
19. ENROLLMENT Suppose the number N (in hundreds) of students projected to attend a high school from 2000 to 2009 can be modeled by the function N(t) = t 2 + 1.5t + 0.5, where t is the number of years since 2000. a. How many students were enrolled in the high school in 2005? b. How many students were enrolled in the high school in 2007?
426  Lesson 74  Polynomials
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. 3
5y + 4x 20. _ 2
21. 21
22. c 4  2c 2 + 1
23. d + 3d c
24. a  a 2
25. 5n 3 + nq 3
x
Example 2
Example 3
Find the degree of each polynomial. 26. 13  4ab + 5a 3b
27. 3x  8
28. 4
29. 17g 2h
30. 10 + 2cd 4  6d 2g
31. 2z 2y 2  7 + 5y 3w 4
Write each polynomial in standard form. Identify the leading coefficient. 32. 5x 2  2 + 3x
33. 8y + 7y 3
34. 4  3c  5c 2
35 4d 4 + 1  d 2
36. 11t + 2t 2  3 + t 5
37. 2 + r  r 3
1 38. _ x  3x 4 + 7
39. 9b 2 + 10b  b 6
2
Example 4
40. FIREWORKS A firework shell is launched two feet from the ground at a speed of 150 feet per second. The height H of the firework shell is modeled by the function H(t) = 16t 2 + 150t + 2, where t is time in seconds. a. How high will the firework be after 3 seconds? b. How high will the firework be after 5 seconds?
B
Classify each polynomial according to its degree and number of terms. 41. 4x  3x 2 + 5
42. 11z 3
43. 9 + y 4
44. 3x 3  7
45. 2z 5  x + 5x  8
2
46. 10t  4t 2 + 6t 3
47. ICE CREAM An ice cream shop is changing the size of their cone. 1 a. If the volume of a cone is the product of _ , π, the square of the radius r, 3 and the height h, write a polynomial that represents the volume.
b. How much will the cone hold if the radius is 1.5 inches and the height is 4 inches? c. If the volume of the cone must be 63 cubic inches and the radius of the cone is 3 inches, how tall is the cone? 48. GEOMETRY Write two expressions for the perimeter and area of the rectangle. 2
4x + 2x  1 2
2x  x + 3
49. GEOMETRY Write a polynomial for the area of the shaded region shown. x
2x
2x 4x
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427
50. PROJECT Rocky and Arturo are designing a rocket for a competition. The top must be coneshaped and the body of the rocket must be cylindrical. The volume of a 1 , π, the height h, and the square of the radius r. The volume cone is the product of _ 3 of a cylinder is the product of π, the height t, and the square of the radius r. a. Write a polynomial that represents the volume of the rocket. b. If the height of the body of the rocket is 8 inches, the height of the top is 6 inches, and the radius is 3 inches, find the volume of the rocket. c. If the height of the body of the rocket is 9 inches, the height of the top is 5 inches, and the radius is 4 inches, find the volume of the rocket.
B
51
MULTIPLE REPRESENTATIONS In this problem, you will explore perimeter and area. a. Geometric Draw three rectangles that each have a perimeter of 400 feet. b. Tabular Record the width and length of each rectangle in a table like the one shown below. Find the area of each rectangle. Rectangle
Length
1
100 ft
2
50 ft
3
75 ft
4
x ft
Width
Area
c. Graphical On a coordinate system, graph the area of rectangle 4 in terms of the length, x. Use the graph to determine the largest area possible. d. Analytical Determine the length and width that produce the largest area.
H.O.T. Problems
Use HigherOrder Thinking Skills
52. ERROR ANALYSIS Chuck and Claudio are writing 2x 2  3 + 5x in standard form. Is either of them correct? Explain your reasoning.
Chuck
Claudio
2x : degree 2 3 : degree 0 5x : degree 1 2x 2 – 5x + 3
2x 2 : degree 2 3 : degree 0 5x : degree 1 2x 2 + 5x – 3
2
53. CHALLENGE Write a polynomial that represents any odd integer if x is an integer. Explain. 54. REASONING Is the following statement sometimes, always, or never true? Explain. A binomial can have a degree of zero. 55. OPEN ENDED Write an example of a cubic trinomial. 56. WRITING IN MATH Explain how to write a polynomial in standard form and how to identify the leading coefficient.
428  Lesson 74  Polynomials
h r t r
SPI 3102.3.6, SPI 3102.1.2, SPI 3102.3.9
Standardized Test Practice 57. Matrices P and Q are given below.
⎡3 2 ⎤ P = ⎢6 9 ⎣1 0 ⎦ What is P  Q?
59. SHORT RESPONSE In a recent poll, 3000 people were asked to pick their favorite baseball team. The accompanying circle graph shows the results of that poll. How many people polled picked the Cubs as their favorite team?
⎡3 2 ⎤ Q = ⎢6 9 ⎣ 4 0⎦
⎡6 9 ⎤ A ⎢8 15 ⎣ 3 0⎦
⎡0 5 ⎤ C ⎢4 3 ⎣5 0 ⎦
⎡ 0 5⎤ B ⎢4 3 ⎣5 0 ⎦
⎡ 6 4⎤ D ⎢ 12 18 ⎣3 0 ⎦
White Sox 1 2 Cubs
58. You have a coupon from The Really Quick Lube Shop for an $8 off oil change this month. An oil change costs $19.95, and a new oil filter costs $4.95. You use the coupon for an oil change and filter. Before adding tax, how much should you pay? F G H J
Red Sox 1 3
60. What value for y satisfies the system of equations below? 2x + y = 19 4x  6y = 2 A B C D
$11.95 $16.90 $24.90 $27.95
5 7 8 10
Spiral Review Express each number in standard notation. (Lesson 73) 10
61. 6 × 10 7
62. 7.2 × 10
64. 7 × 10 6
65. 0.132 × 10 6
Simplify. Assume that no denominator is equal to zero. (Lesson 72) (4m 3c 6) 0 67. a 0(a 4)(a 8) 68. _ mc 70. 12 1
k 4 71. _ 2 8 m p
63. 8.1 × 10 5 66. 1.88 × 10 0 2 6 0
(3f g ) 69. _0
(18f 6g 2) (nq 1) 3 _ 72. (n 4q 8) 1
73. FINANCIAL LITERACY The owners of a new restaurant have hired enough servers to handle 17 tables of customers. The fire marshal has approved the restaurant for a limit of 56 customers. How many twoseat tables and how many fourseat tables should the owners buy? (Lesson 64)
Skills Review Simplify each expression. If not possible, write simplified. (Lesson 15) 74. 7b 2 + 14b  10b
75. 5t + 12t 2  8t
76. 3y 4 + 2y 4 + 2y 5
77. 7h 5  7j 5 + 8k 5
n 2 78. n + _ +_ n
u 79. 2u + _ + u2
3
3
2
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429
MidChapter Quiz
Tennessee Curriculum Standards
Lessons 71 through 74
SPI 3102.1.3
Simplify each expression. (Lesson 71)
Evaluate each product or quotient. Express the results in scientific notation. (Lesson 73)
1. (x 3)(4x 5) 2. (m 2p 5)
19. (2.13 × 10 2)(3 × 10 5)
3
20. (7.5 × 10 6)(2.5 × 10 2)
2 3 3. ⎡⎣(2xy 3) ⎤⎦
7.5 × 10 8 21. _ 4
4. (6ab 3c 4)(3a 2b 3c)
2.5 × 10
5. MULTIPLE CHOICE Express the volume of the solid as a monomial. (Lesson 71)
x
6.6 × 10 5 22. _ 3 2 × 10
Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial.
2
2x 4x
3
4
(Lesson 74)
23. 3y 2  2
A 6x 9
C 8x 24
24. 4t 5 + 3t 2 + t
B 8x 9
D 7x 24
3x 25. _ 5y
26. ax 3 Simplify each expression. Assume that no denominator equals 0. (Lesson 72) 3
0
( )
2xy 7. _
7 4
p t 9. _ 5
2a 4b 3 6. _ 6 c
m n p 8. _ 3 3 m n p
27. 3b 2 28. 2x 3  4x + 1
6x
4 2
r
10. ASTRONOMY Physicists estimate that the number of stars in the universe has an order of magnitude of 10 21. The number of stars in the Milky Way galaxy is around 100 billion. Using orders of magnitude, how many times as many stars are there in the universe as the Milky Way? (Lesson 72)
Express each number in scientific notation. (Lesson 73) 11. 0.00000054
12. 0.0042
13. 234,000
14. 418,000,000
Express each number in standard form. (Lesson 73) 15. 4.1 × 10 3 16. 2.74 × 10 5 17. 3 × 10 9 18. 9.1 × 10 5
430  Chapter 7  MidChapter Quiz
29. POPULATION The table shows the population density for Nevada for various years. (Lesson 74) Year
Years Since 1930
People/ Square Mile
1930
0
0.8
1960
30
2.6
1980
50
7.3
1990
60
10.9
2000
70
18.2
a. The population density d of Nevada from 1930 to 2000 can be modeled by d = 0.005y 2  0.127y + 1, where y represents the number of years since 1930. Identify the type of polynomial for 0.005y 2  0.127y + 1. b. What is the degree of the polynomial? c. Predict the population density of Nevada for 2020. Explain your method. d. Predict the population density of Nevada for 2030. Explain your method.
Algebra Lab
Adding and Subtracting Polynomials Monomials such as 3x and 2x are called like terms because they have the same variable to the same power. When you use algebra tiles, you can recognize like terms because the individual tiles have the same size and shape.
Tennessee Curriculum Standards SPI 3102.3.2 Operate with polynomials and simplify results. Also addresses ✓3102.3.5.
Polynomial Models • Like terms are represented by tiles that have the same shape and size.
Y
• A zero pair may be formed by pairing one tile with its opposite. You can remove or add zero pairs without changing the polynomial.
Y
Y
like terms
Y
0
Y Y
Activity 1 Add Polynomials Use algebra tiles to find (2 x 2  3x + 5) + ( x 2 + 6x  4). Step 1 Model each polynomial.
2x 2  3x + 5
2
Y
Y
2
Y Y Y 1
2
2x
x 2 + 6x  4
YY
+
3x
22
Y
Y
1
Y
Y
1
1
5
+
Y
1
Y 1 1 1 1
x
2
6x
+
+
4
Step 2 Combine like terms and remove zero pairs.
Y
2
Y
Y
2
2
Y
Y
Y
Y
Y
Y
1
Y
Y
1
1
1
1
Y 1 1 1 1
3x
2
+
3x
+
1
Step 3 Write the polynomial for the tiles that remain.
(2x 2  3x + 5) + (x 2 + 6x  4) = 3x 2 + 3x + 1 (continued on the next page) connectED.mcgrawhill.com
431
Algebra Lab
Adding and Subtracting Polynomials Continued Activity 2 Subtract Polynomials Use algebra tiles to find (4x + 5)  (3x + 1). Step 1 Model the polynomial 4x + 5. Y
Y
Y
Y 1
4x
Step 2 To subtract 3x + 1, you must remove 3 red xtiles and 1 yellow 1tile. You can remove the yellow 1tile, but there are no red xtiles. Add 3 zero pairs of xtiles. Then remove the 3 red xtiles.
Y
Y
1
1
Y
Y
1
Y
Y
5
+
Y
1
Y
Y
Y
1
7x
Step 3 Write the polynomial for the tiles that remain. (4x + 5)  (3x + 1) = 7x + 4
1
1
1
1
4
+
Recall that you can subtract a number by adding its additive inverse or opposite. Similarly, you can subtract a polynomial by adding its opposite.
Activity 3 Subtract Polynomials Using Additive Inverse Use algebra tiles to find (4x + 5)  (3x + 1). Step 1 To find the difference of 4x + 5 and 3x + 1, add 4x + 5 and the opposite of 3x + 1.
4x
5
+
4x + 5
Y
Y
Y
The opposite of 3x + 1 is 3x  1.
Y
Y
Y
Y
1
1
1
1
1 3x
+ 1
Step 2 Write the polynomial for the tiles that remain. (4x + 5)  (3x + 1) = 7x + 4. Notice that this is the same answer as in Activity 2.
Model and Analyze Use algebra tiles to find each sum or difference. 1. (x 2 + 5x  2) + (3x 2  2x + 6)
2. (2x 2 + 8x + 1)  (x 2  4x  2)
3. (4x 2 + x)  (x 2 + 5x)
4. WRITING IN MATH Find (4x 2  x + 3)  (2x + 1) using each method from Activity 2 and Activity 3. Illustrate with drawings, and explain in writing how zero pairs are used in each case.
432  Explore 75  Algebra Lab: Adding and Subtracting Polynomials
1
Adding and Subtracting Polynomials Then
Now
Why?
You wrote polynomials in standard form.
1 2
From 2000 to 2003, sales (in millions of dollars) of rap/hiphop music R and country music C in the United States can be modeled by the following equations, where t is the number of years since 2000.
(Lesson 74)
Add polynomials. Subtract polynomials.
R = 132.3t 3 + 624.7t 2  773.6t + 1847.7 C = 3.4t 3 + 8.6t 2  95t + 1532.6 The total music sales T of rap/hiphop music and country music is R + C.
1 Add Polynomials
Adding polynomials involves adding like terms. You can group like terms by using a horizontal or vertical format.
Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. ✔ 3102.3.5 Add, subtract, and multiply polynomials including squaring a binomial. SPI 3102.3.2 Operate with polynomials and simplify results.
Example 1 Add Polynomials a. (2x 2 + 5x  7) + (3  4x 2 + 6x) Horizontal Method
(2x 2 + 5x  7) + (3  4x 2 + 6x) = ⎡⎣2x 2 + (4x 2)⎤⎦ + ⎡⎣5x + 6x⎤⎦ + ⎡⎣7 + 3⎤⎦
Group like terms.
2
= 2x + 11x  4
Combine like terms.
Vertical Method 2x 2 + 5x  7 (__________________ +) 4x 2 + 6x + 3
Align like terms in columns and combine.
2x 2 + 11x  4 b. (3y + y 3  5) + (4y 2  4y + 2y 3 + 8) Horizontal Method
(3y + y 3  5) + (4y 2  4y + 2y 3 + 8) = ⎡⎣y 3 + 2y 3⎤⎦ + 4y 2 + ⎡⎣3y + (4y)⎤⎦ + ⎡⎣(5) + 8⎤⎦
Group like terms.
= 3y 3 + 4y 2  y + 3
Combine like terms.
Vertical Method y 3 + 0y 2 + 3y  5 (_____________________ +) 2y 3 + 4y 2 4y + 8 3y 3 + 4y 2  y + 3
Insert a placeholder to help align the terms. Align and combine like terms.
GuidedPractice 1A. Find (5x 2  3x + 4) + (6x  3x 2  3). 1B. Find (y 4  3y + 7) + (2y 3 + 2y  2y 4  11). connectED.mcgrawhill.com
433
2 Subtract Polynomials
StudyTip Additive Inverse When finding the additive inverse of a polynomial, you are multiplying every term by 1.
Recall that you can subtract a real number by adding its opposite or additive inverse. Similarly, you can subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, write the opposite of each term in the polynomial. (3x 2 + 2x  6) = 3x 2  2x + 6 Additive Inverse
Example 2 Subtract Polynomials Find each difference. a. (3  2x + 2x 2)  (4x  5 + 3x 2) Horizontal Method Subtract 4x  5 + 3x 2 by adding its additive inverse.
(3  2x + 2x 2 )  (4x  5 + 3x 2) = ⎡⎣2x 2 + (3x 2)⎤⎦ + ⎡⎣(2x) + (4x)⎤⎦ + ⎡⎣3 + 5⎤⎦
The additive inverse of 4x  5 + 3x 2 is 4x + 5  3x 2. Group like terms.
= x 2  6x + 8
Combine like terms.
= (3  2x + 2x 2) + (4x + 5  3x 2)
Vertical Method
StudyTip
Align like terms in columns and subtract by adding the additive inverse.
Vertical Method Notice that the polynomials are written in standard form with like terms aligned.
2x 2  2x + 3 () 3x 2 + 4x  5 _______________
2x 2  2x + 3 (+) 3x 2  4x + 5 Add the opposite. _________________ x 2  6x + 8 2) 2) 2 ( ( Thus, 3  2x + 2x  4x  5 + 3x = x  6x + 8. b. (7p + 4p 3  8)  (3p 2 + 2  9p) Horizontal Method Subtract 3p 2 + 2  9p by adding its additive inverse.
(7p + 4p 3  8)  (3p 2 + 2  9p) = (7p + 4p 3  8) + (3p 2  2 + 9p) = ⎡⎣7p + 9p⎤⎦ + 4p 3 + (3p 2) + ⎡⎣(8) + (2)⎤⎦ = 4p 3  3p 2 + 16p  10
The additive inverse of 3p 2 + 2  9p is 3p 2  2 + 9p. Group like terms. Combine like terms.
Vertical Method Align like terms in columns and subtract by adding the additive inverse. 4p 3 + 0p 2 + 7p  8 () 3p 2  9p + 2 _____________________
4p 3 + 0p 2 + 7p  8 (+)  3p 2 + 9p  2 Add the opposite. _____________________ 4p 3  3p 2 + 16p  10 Thus, (7p + 4p 3  8)  (3p 2 + 2  9p) = 4p 3  3p 2 + 16p  10.
GuidedPractice 2A. Find (4x 3  3x 2 + 6x  4)  (2x 3 + x 2  2). 2B. Find (8y  10 + 5y 2)  (7  y 3 + 12y).
434  Lesson 75  Adding and Subtracting Polynomials
RealWorld Example 3 Add and Subtract Polynomials CONSUMER ELECTRONICS An electronics store is starting to track sales of cell phones and digital cameras. The equations below represent the number of cell phones P and the number of digital cameras C sold in m months. P = 7m + 137
C = 4m + 78
a. Write an equation for the monthly sales T of phones and cameras. Add the polynomial for P with the polynomial for C. total sales = cell phone sales + digital camera sales T = 7m + 137 + 4m + 78
Substitution
= 11m + 215
RealWorldLink Sales of digital cameras recently increased by 42% in one year. Sales are expected to increase by at least 15% each year as consumers upgrade their cameras. Source: Big Planet Marketing Company
Combine like terms.
An equation is T = 11m + 215. b. Use the equation to predict the number of cell phones and digital cameras sold in 10 months. T = 11(10) + 215 = 110 + 215
Substitute 10 for m. Simplify.
= 325 Thus, a total of 325 cell phones and digital cameras will be sold in 10 months.
GuidedPractice 3. Use the information above to write an equation that represents the difference in the monthly sales of cell phones and the monthly sales of digital cameras. Use the equation to predict the difference in monthly sales in 24 months.
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–2 Find each sum or difference.
Example 3
1. (6x 3  4) + (2x 3 + 9)
2. (g 3  2g 2 + 5g + 6)  (g 2 + 2g)
3. (4 + 2a 2  2a)  (3a 2  8a + 7)
4. (8y  4y 2) + (3y  9y 2)
5. (4z 3  2z + 8)  (4z 3 + 3z 2  5)
6. (3d 2  8 + 2d) + (4d  12 + d 2)
7 (2c 2 + 6c + 4) + (5c 2  7)
8. (3n 3  5n + n 2)  (8n 2 + 3n 3)
9. VACATION The total number of students T who traveled for spring break consists of two groups: students who flew to their destinations F and students who drove to their destination D. The number (in thousands) of students who flew and the total number of students who flew or drove can be modeled by the following equations, where n is the number of years since 1995. T = 14n + 21
F = 8n + 7
a. Write an equation that models the number of students who drove to their destination for this time period. b. Predict the number of students who will drive to their destination in 2012. c. How many students will drive or fly to their destination in 2015? connectED.mcgrawhill.com
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Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–2 Find each sum or difference.
Example 3
10. (y + 5) + (2y + 4y 2  2)
11 (2x + 3x 2)  (7  8x 2)
12. (3c 3  c + 11)  (c 2 + 2c + 8)
13. (z 2 + z) + (z 2  11)
14. (2x  2y + 1)  (3y + 4x)
15. (4a  5b 2 + 3) + (6  2a + 3b 2)
16. (x 2y  3x 2 + y) + (3y  2x 2y)
17. (8xy + 3x 2  5y) + (4x 2  2y + 6xy)
18. (5n  2p 2 + 2np)  (4p 2 + 4n)
19. (4rxt  8r 2x + x 2)  (6rx 2 + 5rxt  2x 2)
20. (6ab 2 + 2ab) + (3a 2b  4ab + ab 2)
21. (cd 2 + 2cd  4) + (6 + 4cd  2cd 2)
22. PETS From 1999 through 2009, the number of dogs D and the number of cats C (in hundreds) adopted from animal shelters in the United States are modeled by the following equations, where n is the number of years since 1999. D = 2n + 3
C=n+4
a. Write an equation that models the total number T of dogs and cats adopted in hundreds for this time period. b. If this trend continues, how many dogs and cats will be adopted in 2011?
B
Find each sum or difference. 23. (4x + 2y  6z) + (5y  2z + 7x) + (9z  2x  3y) 24. (5a 2  4) + (a 2  2a + 12) + (4a 2  6a + 8) 25. (3c 2  7) + (4c + 7)  (c 2 + 5c  8) 26. (3n 3 + 3n  10)  (4n 2  5n) + (4n 3  3n 2  9n + 4) 27. GEOMETRY Write a polynomial that represents the perimeter of the figure at the right.
2x + 1
_
_
2x + 1
3x  1
2
2
28. PAINTING Kin is painting two walls of her bedroom. The area of one wall can be modeled by 3x 2 + 14, and the area of the other wall can be modeled by 2x  3. What is the total area of the two walls? 29. GEOMETRY The perimeter of the figure at the right is represented by the expression 3x 2  7x + 2. Write a polynomial that represents the measure of the third side.
_
5x + 1 4
x2  x  4
2x 2  10x + 6
30. FOOTBALL The National Football League is divided into two conferences, the American A and the National N. From 2002 through 2009, the total attendance T (in thousands) for both conferences and for the American Conference games can be modeled by the following equations, where x is the number of years since 2002. T = 0.69x 3 + 55.83x 2 + 643.31x + 10,538
A = 3.78x 3 + 58.96x 2 + 265.96x + 5257
Estimate how many people attended a National Conference football game in 2009. 31. GEOMETRY The width of a rectangle is represented by 5x + 2y, and the length is represented by 6y  2x. Write a polynomial that represents the perimeter.
436  Lesson 75  Adding and Subtracting Polynomials
32. GARDENING Candida is planting flowers on the perimeter of a rectangular patio. a. If the perimeter of the patio is 210x and one side measures 32x, find the length of the other side. b. Write a polynomial that represents the area of the rectangular patio. 33. GEOMETRY The sum of the measures of the angles in a triangle is 180°. a. Write an expression to represent the measure of the third angle of the triangle. b. If x = 23, find the measures of the three angles.
(2x  7)°
(4x + 5)°
34. SALES An electronics store estimates that the cost, in dollars, of selling t units of LCD televisions is given by the expression 0.002t 2 + 4t + 400. The revenue from the sales of t LCD televisions is 8t. a. Write a polynomial that represents the profit of selling t units. b. If 750 LCD televisions are sold, how much did the store earn? c. If 575 LCD televisions are sold, how much did the store earn? 35 CAR RENTAL The cost to rent a car for a day is $15 plus $0.15 for each mile driven. a. Write a polynomial that represents the cost of renting a car for m miles. b. If a car is driven 145 miles, how much would it cost to rent? c. If a car is driven 105 miles each day for four days, how much would it cost to rent a car? d. If a car is driven 220 miles each day for seven days, how much would it cost to rent a car?
H.O.T. Problems C
Use HigherOrder Thinking Skills
36. ERROR ANALYSIS Cheyenne and Sebastian are finding (2x 2  x)  (3x + 3x 2  2). Is either of them correct? Explain your reasoning.
Cheyenne
Sebastian
(2x 2 – x) – (3x + 3x 2 – 2) = (2x 2 – x) + (3x + 3x 2 – 2) = 5x 2 – 4x – 2
(2x 2 – x) – (3x + 3x 2 – 2) = (2x 2 – x) + (3x  3x 2 – 2) = –x 2 – 4x – 2
37. OPEN ENDED Write two trinomials with a difference of 2x 3  7x + 8. 38. CHALLENGE Write a polynomial that represents the sum of an odd integer 2n + 1 and the next two consecutive odd integers. 39. REASONING Find a counterexample to the following statement. The order in which polynomials are subtracted does not matter. 40. OPEN ENDED Write three trinomials with a sum of 4x 4 + 3x 2. 41.
E WRITING IN MATH Describe how to add and subtract polynomials using both the vertical and horizontal formats. Which one do you think is easier? Why? connectED.mcgrawhill.com
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SPI 3102.3.2, SPI 3108.1.4, SPI 3102.3.9
Standardized Test Practice 42. Three consecutive integers can be represented by x, x + 1, and x + 2. What is the sum of these three integers? A x(x + 1)(x + 2)
C 3x + 3
B x3 + 3
D x+3
45. Which ordered pair is in the solution set of the system of inequalities shown in the graph? y
43. SHORT RESPONSE What is the perimeter of a square with sides that measure 2x + 3 units?
0
44. Jim cuts a board in the shape of a regular hexagon and pounds in a nail at each vertex, as shown. How many rubber bands will he need to stretch a rubber band across every possible pair of nails? F 15
G 14
H 12
A (3, 0)
C (5, 0)
B (0, 3)
D (0, 5)
x
J 9
Spiral Review Find the degree of each polynomial. (Lesson 74) 46. 6b 4
47. 10t
48. 5g 2h
49. 7np 4
50. 25
51. t 3 + 6u
52. 2 + 3ab 3  a 2b + 4a 6
53. 6  v 4 + v 2z 3 + 6v 3
54. POPULATION The 2008 population of North Carolina’s Beaufort County was approximately 46,000. Express this number in scientific notation. (Lesson 73) 55. JOBS Kimi received an offer for a new job. She wants to compare the offer with her current job. What is total amount of sales that Kimi must get each month to make the same income at either job? (Lesson 62)
New Offer $600/mo 2% commission Current Job $1000/mo 1.5% commission
Determine whether each sequence is an arithmetic sequence. If it is, state the common difference. (Lesson 35) 56. 24, 16, 8, 0, …
1 _ 57. 3_ , 6 1 , 13, 26, …
58. 7, 6, 5, 4, …
59. 10, 12, 15, 18, …
60. 15, 11, 7, 3, …
61. 0.3, 0.2, 0.7, 1.2, …
4
2
Skills Review Simplify. (Lesson 71) 62. t(t 5)(t 7)
63. n 3(n 2)(2n 3)
64. (5t 5v 2)(10t 3v 4)
65. (8u 4z 5)(5uz 4)
66. ⎡⎣(3) 2⎤⎦ 3
67. ⎡⎣(2) 3⎤⎦ 2
2 68. (2m 4k 3) (3mk 2) 3
69. (6xy 2) (2x 2y 2z 2)
438  Lesson 75  Adding and Subtracting Polynomials
2
3
Multiplying a Polynomial by a Monomial Then
Now
Why?
You multiplied monomials.
1 2
Charmaine Brooks is opening a fitness club. She tells the contractor that the length of the fitness room should be three times the width plus 8 feet.
(Lesson 71)
Multiply a polynomial by a monomial. Solve equations involving the products of monomials and polynomials.
To cover the floor with mats for exercise classes, Ms. Brooks needs to know the area of the floor. So she multiplies the width times the length, w (3w + 8).
1 Polynomial Multiplied by Monomial
To find the product of a polynomial and a monomial, you can use the Distributive Property.
Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. SPI 3102.3.2 Operate with polynomials and simplify results.
Example 1 Multiply a Polynomial by a Monomial Find 3x 2(7x 2  x + 4). Horizontal Method 3x 2(7x 2  x + 4) 2(
= 3x 7x
2)
Original expression 2)
2)
 (3x (x) + (3x (4)
= 21x  (3x 4
3)
+ (12x
2)
= 21x 4 + 3x 3  12x 2
Distributive Property Multiply. Simplify.
Vertical Method 7x 2  x + 4 (×) 3x 2 __________________ 21x 4 + 3x 3  12x 2
Distributive Property Multiply.
GuidedPractice Find each product. 1A. 5a 2(4a 2 + 2a  7)
1B. 6d 3(3d 4  2d 3  d + 9)
We can use this same method more than once to simplify large expressions.
Example 2 Simplify Expressions Simplify 2p(4p 2 + 5p)  5(2p 2 + 20). 2p(4p 2 + 5p)  5(2p 2 + 20)
Original expression
= (2p)(4p 2) + (2p)(5p) + (5)(2p 2) + (5)(20)
Distributive Property
= 8p 3 + 10p 2  10p 2  100
Multiply.
= 8p 3 + (10p 2  10p 2)  100
Commutative and Associative Properties
= 8p 3  100
Combine like terms. connectED.mcgrawhill.com
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GuidedPractice Simplify each expression. 2A. 3(5x 2 + 2x  4)  x(7x 2 + 2x  3)
2B. 15t(10y 3t 5 + 5y 2t)  2y(yt 2 + 4y 2)
We can use the Distributive Property to multiply monomials by polynomials and solve real world problems. SPI 3102.1.3
Test Example 3 GRIDDED RESPONSE The theme for a school dance is “Solid Gold.” For one decoration, Kana is covering a trapezoidshaped piece of poster board with metallic gold paper to look like a bar of gold. If the height of the poster board is 18 inches, how much metallic paper will Kana need in square inches?
h+1
h 2h + 4
TestTakingTip
Read the Test Item
Formulas Many standardized tests provide formula sheets with commonly used formulas. If you are unsure of the correct formula, check the sheet before beginning to solve the problem.
The question is asking you to find the area of the trapezoid with a height of h and bases of h + 1 and 2h + 4. Solve the Test Item Write an equation to represent the area of the trapezoid. Let b 1 = h + 1, let b 2 = 2h + 4 and let h = height of the trapezoid. 1 h(b 1 + b 2) A=_
2 1 =_ h[(h + 1) + (2h+ 4)] 2 1 =_ h(3h + 5) 2 3 2 _ =_ h + 5h 2 2 3 5 =_ (18) 2 + _ (18) 2 2
= 531
Area of a trapezoid b 1 = h + 1 and b 2 = 2h + 4 Add and simplify. Distributive Property h = 18 Simplify.
5 3 1
Kana will need 531 square inches of metallic paper. Grid in your response of 531.
GuidedPractice
RealWorldLink In a recent year, the pet supply business hit an estimated $7.05 billion in sales. This business ranges from gourmet food to rhinestone tiaras, pearl collars, and cashmere coats. Source: Entrepreneur Magazine
3. Kachima is making triangular bandanas for the dogs and cats in her pet club. The base of the bandana is the length of the collar with 4 inches added to each 1 of the collar length. end to tie it on. The height is _ 2
A. If Kachima’s dog has a collar length of 12 inches, how much fabric does she need in square inches? B. If Kachima makes a bandana for her friend’s cat with a 6inch collar, how much fabric does Kachima need in square inches?
440  Lesson 76  Multiplying a Polynomial by a Monomial
2 Solve Equations with Polynomial Expressions
We can use the Distributive Property to solve equations that involve the products of monomials and polynomials.
RealWorld Example 4 Equations with Polynomials on Both Sides
StudyTip Combining Like Terms When simplifying a long expression, it may be helpful to put a circle around one set of like terms, a rectangle around another set, a triangle around another set, and so on.
Solve 2a(5a  2) + 3a(2a + 6) + 8 = a(4a + 1) + 2a(6a  4) + 50. 2a(5a  2) + 3a(2a + 6) + 8 = a(4a + 1) + 2a(6a  4) + 50 10a 2  4a + 6a 2 + 18a + 8 = 4a 2 + a + 12a 2  8a + 50 16a 2 + 14a + 8 = 16a 2  7a + 50 14a + 8 = 7a + 50 21a + 8 = 50 21a = 42 a=2 CHECK
Original equation Distributive Property Combine like terms. Subtract 16a 2 from each side. Add 7a to each side. Subtract 8 from each side. Divide each side by 21.
2a(5a  2) + 3a(2a + 6) + 8 = a(4a + 1) + 2a(6a  4) + 50 2(2)[5(2)  2] + 3(2)[2(2) + 6] + 8 2[4(2) + 1] + 2(2)[6(2)  4] + 50 Simplify. 4(8) + 6(10) + 8 2(9) + 4(8) + 50 32 + 60 + 8 18 + 32 + 50 Multiply. 100 = 100 Add and subtract.
GuidedPractice Solve each equation. 4A. 2x(x + 4) + 7 = (x + 8) + 2x(x + 1) + 12 4B. d(d + 3)  d(d  4) = 9d  16
Check Your Understanding Example 1
Example 2
= StepbyStep Solutions begin on page R12.
Find each product. 1. 5w(3w 2 + 2w  4)
2. 6g 2(3g 3 + 4g 2 + 10g  1)
3. 4km 2(8km 2 + 2k 2m + 5k)
4. 3p 4r 3(2p 2r 4  6p 6r 3  5)
5 2ab(7a 4b 2 + a 5b  2a)
6. c 2d 3(5cd 7  3c 3d 2  4d 3)
Simplify each expression. 7. t(4t 2 + 15t + 4)  4(3t  1)
8. x(3x 2 + 4) + 2(7x  3)
9. 2d(d 3c 2  4dc 2 + 2d 2c) + c 2(dc 2  3d 4) 10. 5w 2(8w 2x  11wx 2) + 6x(9wx 4  4w  3x 2) Example 3
11. GRIDDED RESPONSE Marlene is buying a new plasma television. The height of the screen of the television is one half the width plus 5 inches. The width is 30 inches. Find the height of the screen in inches.
Example 4
Solve each equation. 12. 6(11  2c) = 7(2  2c)
13. t(2t + 3) + 20 = 2t(t  3)
14. 2(w + 1) + w = 7  4w
15. 3(y  2) + 2y = 4y + 14
16. a(a + 3) + a(a  6) + 35 = a(a  5) + a(a + 7) 17. n(n  4) + n(n + 8) = n(n  13) + n(n + 1) + 16 connectED.mcgrawhill.com
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Practice and Problem Solving Example 1
Example 2
Extra Practice begins on page 815.
Find each product. 18. b(b 2  12b + 1)
19. f(f 2 + 2f + 25)
20. 3m 3(2m 3  12m 2 + 2m + 25)
21. 2j 2(5j 3  15j 2 + 2j + 2)
22. 2pr 2(2pr + 5p 2r  15p)
23. 4t 3u(2t 2u 2  10tu 4 + 2)
Simplify each expression. 24. 3(5x 2 + 2x + 9) + x(2x  3)
25. a(8a 2 + 2a + 4) + 3(6a 2  4)
26. 4d(5d 2  12) + 7(d + 5)
27. 9g(2g + g 2) + 3(g 2 + 4)
28. 2j(7j 2k 2 + jk 2 + 5k)  9k(2j 2k 2 + 2k 2 + 3j) 29. 4n(2n 3p 2  3np 2 + 5n) + 4p(6n 2p  2np 2 + 3p) Example 3
30. DAMS A new dam being built has the shape of a trapezoid. The base at the bottom of the dam is 2 times the height. 1 times the height The base at the top of the dam is _ 5 minus 30 feet. a. Write an expression to find the area of the trapezoidal cross section of the dam. b. If the height of the dam is 180 feet, find the area of this cross section.
Example 4
Solve each equation. 31 7(t 2 + 5t  9) + t = t(7t  2) + 13 32. w(4w + 6) + 2w = 2(2w 2 + 7w  3) 33. 5(4z + 6)  2(z  4) = 7z(z + 4)  z(7z  2)  48 34. 9c(c  11) + 10(5c  3) = 3c(c + 5) + c(6c  3)  30 35. 2f(5f  2)  10(f 2  3f + 6) = 8f(f + 4) + 4(2f 2  7f) 36. 2k(3k + 4) + 6(k 2 + 10) = k(4k + 8)  2k(2k + 5)
B
Simplify each expression. 2 2 37. _ np (30p 2 + 9n 2p  12)
3 2 38. _ r t(10r 3 + 5rt 3 + 15t 2)
3
2
3
5
2
2
39. 5q w (4q + 7w) + 4qw (7q w + 2q)  3qw(3q 2w 2 + 9) 40. x 2z(2z 2 + 4xz 3) + xz 2(xz + 5x 3z) + x 2z 3(3x 2z + 4xz) 41. PARKING A parking garage charges $30 per month plus $0.50 per daytime hour and $0.25 per hour during nights and weekends. Suppose Trent parks in the garage for 47 hours in January and h of those are night and weekend hours.
h+4
a. Find an expression for Trent’s January bill. b. Find the cost if Trent had 12 hours of night and weekend hours.
h 3h + 1
42. PETS Che is building a dog house for his new puppy. The upper face of the dog house is a trapezoid. If the height of the trapezoid is 12 inches, find the area of the face of this piece of the dog house.
442  Lesson 76  Multiplying a Polynomial by a Monomial
43 TENNIS The tennis club is building a new tennis court with a path around it. a. Write an expression for the area of the tennis court.
2.5x
x+6
x
b. Write an expression for the area of the path. 1.5x + 24
c. If x = 36 feet, what is the perimeter of the outside of the path?
C
44.
MULTIPLE REPRESENTATIONS In this problem, you will investigate the degree of the product of a monomial and a polynomial. a. Tabular Write three monomials of different degrees and three polynomials of different degrees. Determine the degree of each monomial and polynomial. Multiply the monomials by the polynomials. Determine the degree of each product. Record your results in a table like the one shown below. Monomial
Degree
Polynomial
Degree
Product of Monomial and Polynomial
Degree
b. Verbal Make a conjecture about the degree of the product of a monomial and a polynomial. What is the degree of the product of a monomial of degree a and a polynomial of degree b?
H.O.T. Problems
Use HigherOrder Thinking Skills
45. ERROR ANALYSIS Pearl and Ted both worked on this problem. Is either of them correct? Explain your reasoning.
Ted
Pearl 2
2
2x (3x + 4x + 2) 6x 4 + 8x 2 + 4x 2 6x 4 + 12x 2
2
2x (3x 2 + 4x + 2) 6x 4 + 8x 3 + 4x 2
46. CHALLENGE Find p such that 3x p(4x 2p + 3 + 2x 3p  2) = 12x 12 + 6x 10. 47. CHALLENGE Simplify 4x 3y 2(2x 5y 4 + 6x 7y 6  4x 0y 2). 48. REASONING Is there a value for x that makes the statement (x + 2)2 = x 2 + 2 2 true? If so, find a value for x. Explain your reasoning. 49. OPEN ENDED Write a monomial and a polynomial using n as the variable. Find their product. 50.
E
WRITING IN MATH Describe the steps to multiply a polynomial by a monomial. connectED.mcgrawhill.com
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SPI 3102.1.4, SPI 3102.1.3, SPI 3108.4.6, SPI 3102.3.6
Standardized Test Practice 51. Every week a store sells j jeans and t Tshirts. The store makes $8 for each Tshirt and $12 for each pair of jeans. Which of the following expressions represents the total amount of money, in dollars, the store makes every week? A 8j + 12t B 12j + 8t
53. GEOMETRY A triangle has sides of length 5 inches and 8.5 inches. Which of the following cannot be the length of the third side? A B C D
C 20(j + t) D 96jt
52. If a = 5x + 7y and b = 2y  3x, what is a + b? F 2x  9y G 3y + 4x
H 2x + 9y J 2x  5y
3.5 inches 4 inches 5.5 inches 12 inches
54. SHORT RESPONSE Write an equation in which x varies directly as the cube of y and inversely as the square of z.
Spiral Review Find each sum or difference. (Lesson 75) 55. (2x 2  7) + (8  5x 2)
56. (3z 2 + 2z  1) + (z 2  6)
57. (2a  4a 2 + 1)  (5a 2  2a  6)
58. (a 3  3a 2 + 4)  (4a 2 + 7)
59. (2ab  3a + 4b) + (5a + 4ab)
60. (8c 3  3c 2 + c  2)  (3c 3 + 9)
Find the degree of each polynomial. (Lesson 74) 61. 12y
62. 10
63. 2x 2  5
64. 9a  8a 3 + 6
65. 7b 2c 3
66. 3p 4r 5t 2
67. TRAVEL In 2003, about 9.5 million people took cruises. Between 2003 and 2008, the number increased by about 740,000 each year. Write the pointslope form of an equation to find the total number of people y taking a cruise for any year x. Estimate the number of people who will take a cruise in 2010. (Lesson 43) Write an equation in function notation for each relation. (Lesson 36) y
68.
0
y
69.
x
0
x
Skills Review Simplify. (Lesson 71) 70. b(b 2)(b 3)
71. 2y(3y 2)
72. y 4(2y 3)
73. 3z 3(5z 4 + 2z)
74. 2m(4m 4)  3(5m 3)
75. 4p 2(2p 3) + 2p 4(5p 6)
444  Lesson 76  Multiplying a Polynomial by a Monomial
Algebra Lab
Multiplying Polynomials You can use algebra tiles to find the product of two binomials. Tennessee Curriculum Standards SPI 3102.3.2 Operate with polynomials and simplify results. Also addresses ✓3102.3.5 and ✓3102.3.7.
Activity 1 Multiply Binomials Use algebra tiles to find (x + 3)(x + 4). The rectangle will have a width of x + 3 and a length of x + 4. Use algebra tiles to mark off the dimensions on a product mat. Then complete the rectangle with algebra tiles. x
1
1
1
x
1
x
x
Y
1
1
1
1
1
1
Y Y Y
1 2
1
1
1
Y Y Y Y 1 1
1 1
1 1
1 1
1
1
1
1
The rectangle consists of 1 blue x 2tile, 7 green xtiles, and 12 yellow 1tiles. The area of the rectangle is x 2 + 7x + 12. So, (x + 3)(x + 4) = x 2 + 7x + 12.
Activity 2 Multiply Binomials Use algebra tiles to find (x  2)(x  5). Step 1 The rectangle will have a width of x  2 and a length of x  5. Use algebra tiles to mark off the dimensions on a product mat. Then begin to make the rectangle with algebra tiles.
x x
x
1
1
1
1
Step 2 Determine whether to use 10 yellow 1tiles or 10 red 1tiles to complete the rectangle. The area of each yellow tile is the product of 1 and 1. Fill in the space with 10 yellow 1tiles to complete the rectangle. 2
x
1 1 1 1 1
The rectangle consists of 1 blue x tile, 7 red xtiles, and 10 yellow 1tiles. The area of the rectangle is x 2  7x + 10. So, (x  2)(x  5) = x 2  7x + 10.
Y
1 1 1 1 1 2
Y Y Y Y Y
Y Y
x5
x2
Y
2
Y Y
Y Y Y Y Y 1 1
1 1
1 1
1 1
1 1
(continued on the next page) connectED.mcgrawhill.com
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Algebra Lab
Multiplying Polynomials Continued Activity 3 Multiply Binomials Use algebra tiles to find (x  4)(2x + 3). x
Step 1 The rectangle will have a width of x  4 and a length of 2x + 3. Use algebra tiles to mark off the dimensions on a product mat. Then begin to make the rectangle with algebra tiles.
x
x
1 1 1
x
x
1
1
1 1 1
1 1 1
Y
2
Y
1 1 1
2
Y Y Y
Y Y Y Y 2x + 3
Step 2 Determine what color xtiles and what color 1tiles to use to complete the rectangle. The area of each red xtile is the product of x and 1. The area of each red 1tile is represented by the product of 1 and 1 or 1.
Y x4
Complete the rectangle with 4 red xtiles and 12 red 1tiles.
Step 3 Rearrange the tiles to simplify the polynomial you have formed. Notice that a 3 zero pair are formed by three positive and three negative xtiles.
x
Y
2
Y
2
2
Y
Y Y Y Y
Y
2
Y Y Y
Y Y Y Y
Y
1 1 1 1
1 1 1 1
Y Y
Y
1 1 1 1
Y
2
There are 2 blue x tiles, 5 red xtiles, and 12 red 1tiles left. In simplest form, (x  4)(2x + 3) = 2x 2  5x  12.
Y Y Y Y Y
1
1
1
1
1
1
1
1
1
1
1
1
Model and Analyze Use algebra tiles to find each product. 1. (x + 1)(x + 4)
2. (x  3)(x  2)
3. (x + 5)(x  1)
4. (x + 2)(2x + 3)
5. (x  1)(2x  1)
6. (x + 4)(2x  5)
Is each statement true or false? Justify your answer with a drawing of algebra tiles. 7. (x  4)(x  2) = x 2  6x + 8
8. (x + 3)(x + 5) = x 2 + 15
9. WRITING IN MATH You can also use the Distributive Property to find the product of two binomials. The figure at the right shows the model for (x + 4)(x + 5) separated into four parts. Write a sentence or two explaining how this model shows the use of the Distributive Property.
446  Explore 77  Algebra Lab: Multiplying Polynomials
Y
2
Y Y Y Y
Y Y Y Y Y
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
Multiplying Polynomials Then
Now
Why?
You multiplied polynomials by monomials.
1
Multiply polynomials by using the Distributive Property.
2
Multiply binomials by using the FOIL method.
Bodyboards, which are used to ride waves, are made of foam and are more rectangular than surfboards. A bodyboard’s dimensions are determined by the height and skill level of the user.
(Lesson 76)
The length of Ann’s bodyboard should be Ann’s height h minus 32 inches or h  32. The board’s width should be half of Ann’s height plus 11 inches or
_1 h + 11. To approximate the area of the bodyboard, 2
1 you need to find (h  32) _ h + 11 .
NewVocabulary FOIL method quadratic expression
1
(2
)
Multiply Binomials To multiply two binomials such as h  32 and _12 h + 11, the Distributive Property is used. Binomials can be multiplied horizontally or vertically.
Example 1 The Distributive Property Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. ✔ 3102.3.5 Add, subtract, and multiply polynomials including squaring a binomial. SPI 3102.3.2 Operate with polynomials and simplify results. Also addresses ✓3102.3.7.
Find each product. a. (2x + 3)(x + 5) Vertical Method Multiply by 5.
Multiply by x.
Combine like terms.
2x + 3 (×) x + 5 _________ 10x + 15
2x + 3 (×) x+5 ____________ 10x + 15 2x 2 + 3x _____________
2x + 3 (×) x+5 _____________ 10x + 15 2x 2 + 3x _____________
x(2x + 3) = 2x 2 + 3x
2x 2 + 13x + 15
5(2x + 3) = 10x + 15 Horizontal Method
(2x + 3)(x + 5) = 2x(x + 5) + 3(x + 5) = 2x 2 + 10x + 3x + 15 = 2x 2 + 13x + 15
Rewrite as the sum of two products. Distributive Property Combine like terms.
b. (x  2)(3x + 4) Vertical Method Multiply by 4.
Multiply by 3x.
x2 (×) 3x +4 _________ 4x  8
x2 (×) 3x +4 __________ 4x  8 3x 2  6x ____________ 3x(x  2) = 3x 2  6x
4(x  2) = 4x  8
Horizontal Method (x  2)(3x + 4) = x(3x + 4)  2(3x + 4) 2 = 3x + 4x  6x  8 = 3x 2  2x  8
Combine like terms. x2 (×) 3x +4 __________ 4x  8 3x 2  6x ____________ 3x 2  2x  8
Rewrite as the difference of two products. Distributive Property Combine like terms. connectED.mcgrawhill.com
447
GuidedPractice 1A. (3m + 4)(m + 5)
1B. (5y  2)(y + 8)
A shortcut version of the Distributive Property for multiplying binomials is called the FOIL method.
KeyConcept FOIL Method To multiply two binomials, find the sum of the products of F the First terms, O the Outer terms, I the Inner terms, L and the Last terms.
Words Example
F
ReadingMath Polynomials as Factors The expression (x + 4)(x  2) is read the quantity x plus 4 times the quantity x minus 2.
L
Product of First Terms
(x + 4)(x  2) = (x)(x) I O
+
(x)(2)
=
x 2  2x + 4x  8
=
x 2 + 2x  8
Product of Last Terms
Product of Inner Terms
Product of Outer Terms
+
(4)(x)
+
(4)(2)
Example 2 FOIL Method Find each product. a. (2y  7)(3y + 5) F
L
(2y  7)(3y + 5) = (2y)(3y) + (2y)(5) + (7)(3y) + (7)(5) = 6y 2 + 10y  21y  35 I = 6y 2  11y  35
FOIL method Multiply. Combine like terms.
O
b. (4a  5)(2a  9) (4a  5)(2a  9) = (4a)(2a) + (4a)(9) + (5)(2a) + (5)(9) = 8a 2  36a  10a + 45 = 8a 2  46a + 45
FOIL method Multiply. Combine like terms.
GuidedPractice 2A. (x + 3)(x  4)
2B. (4b  5)(3b + 2)
2C. (2y  5)(y  6)
2D. (5a + 2)(3a  4)
Notice that when two linear expressions are multiplied, the result is a quadratic expression. A quadratic expression is an expression in one variable with a degree of 2. When three linear expressions are multiplied, the result has a degree of 3. The FOIL method can be used to find an expression that represents the area of a rectangular object when the lengths of the sides are given as binomials.
448  Lesson 77  Multiplying Polynomials
RealWorld Example 3 FOIL Method SWIMMING POOL A contractor is building a deck around a rectangular swimming pool. The deck is x feet from every side of the pool. Write an expression for the total area of the pool and deck.
RealWorldLink The cost of a swimming pool depends on many factors, including the size of the pool, whether the pool is an aboveground or an inground pool, and the material used. Source: American Dream Homes
15 ft x
20 ft
Understand We need to find an expression for the total area of the pool and deck.
x
Plan Use the formula for the area of a rectangle and determine the length and width of the pool with the deck. Solve Since the deck is the same distance from every side of the pool, the length and width of the pool are 2x longer. So, the length can be represented by 2x + 20 and the width can be represented by 2x + 15. Area = length · width
Area of a rectangle
= (2x + 20)(2x + 15)
Substitution
= (2x)(2x) + (2x)(15) + (20)(2x) + (20)(15)
FOIL Method
2
Multiply.
2
Combine like terms.
= 4x + 30x + 40x + 300 = 4x + 70x + 300
So, the total area of the deck and pool is 4x 2 + 70x + 300. Check Choose a value for x. Substitute this value into (2x + 20)(2x + 15) and 4x 2 + 70x + 300. The result should be the same for both expressions.
GuidedPractice 3. If the pool is 25 feet long and 20 feet wide, find the area of the pool and deck.
2 Multiply Polynomials
The Distributive Property can also be used to multiply
any two polynomials.
Example 4 The Distributive Property Find each product.
StudyTip Multiplying Polynomials If a polynomial with c terms and a polynomial with d terms are multiplied together, there will be c · d terms before simplifying. In Example 4a, there are 2 · 3 or 6 terms before simplifying.
a. (6x + 5)(2x 2  3x  5) (6x + 5)(2x 2  3x  5) = 6x(2x 2  3x  5) + 5(2x 2  3x  5) = 12x 3  18x 2  30x + 10x 2  15x  25 = 12x 3  8x 2  45x  25
Distributive Property Multiply. Combine like terms.
b. (2y + 3y  1)(3y  5y + 2) (2y 2 + 3y  1)(3y 2  5y + 2) = 2y 2(3y 2  5y + 2) + 3y(3y 2  5y + 2)  1(3y 2  5y + 2) 2
2
4
3
2
3
2
2
= 6y  10y + 4y + 9y  15y + 6y  3y + 5y  2 4
3
2
= 6y  y  14y + 11y  2
Distributive Property Multiply. Combine like terms.
GuidedPractice 4A. (3x  5)(2x 2 + 7x  8)
4B. (m 2 + 2m  3)(4m 2  7m + 5) connectED.mcgrawhill.com
449
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–2 Find each product.
Example 3
Example 4
1. (x + 5)(x + 2)
2. (y  2)(y + 4)
3. (b  7)(b + 3)
4. (4n + 3)(n + 9)
5. (8h  1)(2h  3)
6. (2a + 9)(5a  6)
7. FRAME Hugo is designing a frame to surround the picture shown at the right. The frame is the same distance all the way around. Write an expression that represents the total area of the picture and frame.
16 in.
Find each product.
20 in.
x x
8. (2a  9)(3a + 4a  4) 2
9. (4y 2  3)(4y 2 + 7y + 2) 10. (x 2  4x + 5)(5x 2 + 3x  4) 11. (2n 2 + 3n  6)(5n 2  2n  8)
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–2 Find each product. 12. (3c  5)(c + 3)
13. (g + 10)(2g  5)
14. (6a + 5)(5a + 3)
15 (4x + 1)(6x + 3)
16. (5y  4)(3y  1)
17. (6d  5)(4d  7)
18. (3m + 5)(2m + 3)
19. (7n  6)(7n  6)
20. (12t  5)(12t + 5)
21. (5r + 7)(5r  7)
22. (8w + 4x)(5w  6x)
23. (11z  5y)(3z + 2y)
Example 3
24. GARDEN A walkway surrounds a rectangular garden. The width of the garden is 8 feet, and the length is 6 feet. The width x of the walkway around the garden is the same on every side. Write an expression that represents the total area of the garden and walkway.
Example 4
Find each product.
B
25. (2y  11)(y 2  3y + 2)
26. (4a + 7)(9a 2 + 2a  7)
27. (m 2  5m + 4)(m 2 + 7m  3)
28. (x 2 + 5x  1)(5x 2  6x + 1)
29. (3b 3  4b  7)(2b 2  b  9)
30. (6z 2  5z  2)(3z 3  2z  4)
Simplify. 31. (m + 2)⎡⎣(m 2 + 3m  6) + (m 2  2m + 4)⎤⎦ 32. ⎡⎣(t 2 + 3t  8)  (t 2  2t + 6)⎤⎦(t  4) GEOMETRY Find an expression to represent the area of each shaded region. 33.
34. 2x + 3
4x + 1
x+1 3x + 2 5x
450  Lesson 77  Multiplying Polynomials
2x  3
35 VOLLEYBALL The dimensions of a sand volleyball court are represented by a width of 6y  5 feet and a length of 3y + 4 feet. a. Write an expression that represents the area of the court. b. The length of a sand volleyball court is 31 feet. Find the area of the court. 36. GEOMETRY Write an expression for the area of a triangle with a base of 2x + 3 and a height of 3x  1. Find each product. 37. (a  2b)2
38. (3c + 4d)2
39. (x  5y)2
40. (2r  3t)3
41. (5g + 2h)3
42. (4y + 3z)(4y  3z)2
43. CONSTRUCTION A sandbox kit allows you to build a square sandbox or a rectangular sandbox as shown.
Ym Ym
a. What are the possible values of x? Explain. Y
b. Which shape has the greater area?
Ym
c. What is the difference in areas between the two?
C
44.
MULTIPLE REPRESENTATIONS In this problem, you will investigate the square of a sum. a. Tabular Copy and complete the table for each sum. Expression
(Expression)2
x+5 3y + 1 z+q
b. Verbal Make a conjecture about the terms of the square of a sum. c. Symbolic For a sum of the form a + b, write an expression for the square of the sum.
H.O.T. Problems
Use HigherOrder Thinking Skills
45. REASONING Determine if the following statement is sometimes, always, or never true. Explain your reasoning. The FOIL method can be used to multiply a binomial and a trinomial. 46. CHALLENGE Find (x m + x p)(x m  1  x 1  p + x p). 47. OPEN ENDED Write a binomial and a trinomial involving a single variable. Then find their product. 48. REASONING Compare and contrast the procedure used to multiply a trinomial by a binomial using the vertical method with the procedure used to multiply a threedigit number by a twodigit number. 49. WRITING IN MATH Summarize the methods that can be used to multiply polynomials. connectED.mcgrawhill.com
451
SPI 3102.3.2, SPI 3108.3.2, SPI 3102.2.3, SPI 3102.1.6
Standardized Test Practice 50. What is the product of 2x  5 and 3x + 4? A 5x  1 B 6x 2  7x  20
1
C 6x 2  20
23
5 4 3 2 1
D 6x 2 + 7x  20 51. Which statement is correct about the symmetry of this design? y
(1, 8)
(1, 4) (1, 4)
(3, 4)
(3, 6)
F The design is symmetrical only about the yaxis. G The design is symmetrical only about the xaxis.
J The design has no symmetry.
1
2
3
4
5
53. SHORT RESPONSE For a science project, Jodi selected three bean plants of equal height. Then, for five days, she measured their heights in centimeters and plotted the values on the graph below.
(3, 6)
H The design is symmetrical about both the y and the xaxes.
5
C R D T
x
0
0
A P B Q
Height (cm)
(1, 8)
(3, 4)
52. Which point on the number line represents a number that, when cubed, will result in a number greater than itself?
10 9 8 7 6 5 4 3 2 1
y
Bean Plant Growth
(5, 7)
(1, 1)
1 2 3 4 5 6 x
0
Day
She drew a line of best fit on the graph. What is the slope of the line that she drew?
Spiral Review 54. SAVINGS Carrie has $6000 to invest. She puts x dollars of this money into a savings account that earns 2% interest per year. She uses the rest of the money to purchase a certificate of deposit that earns 4% interest. Write an equation for the amount of money that Carrie will have in one year. (Lesson 76) Find each sum or difference. (Lesson 75) 55. (7a 2  5) + (3a 2 + 10)
56. (8n  2n 2) + (4n  6n 2)
57. (4 + n 3 + 3n 2) + (2n 3  9n 2 + 6)
58. (4u 2  9 + 2u) + (6u + 14 + 2u 2)
59. (b + 4) + (c + 3b  2)
60. (3a 3  6a)  (3a 3 + 5a)
61. (4m 3  m + 10)  (3m 3 + 3m 2  7)
62. (3a + 4ab + 3b)  (2b + 5a + 8ab)
Skills Review Simplify. (Lesson 71) 3 4 63. (2t 4)  3(2t 3)
64. (3h 2)3  2(h 3)2
452  Lesson 77  Multiplying Polynomials
2
3
65. 2(5y 3) + (3y 3)
2
2
66. 3(6n 4) + (2n 2)
Special Products Then
Now
Why?
You multiplied binomials by using the FOIL method.
1 2
Colby wants to attach a dartboard to a square piece of corkboard. If the radius of the dartboard is r + 12, how large does the square corkboard need to be?
(Lesson 77)
Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials.
Find squares of sums and differences. Find the product of a sum and a difference.
Colby knows that the diameter of the dartboard is 2(r + 12) or 2r + 24. Each side of the square also measures 2r + 24. To find how much corkboard is needed, Colby must find the area of the square: A = (2r + 24) 2.
1 Squares of Sums and Differences
Some pairs of binomials, such as squares like (2r + 24) 2, have products that follow a specific pattern. Using the pattern can make multiplying easier. The square of a sum, (a + b) 2 or (a + b)(a + b), is one of those products. a+b a
✔ 3102.3.5 Add, subtract, and multiply polynomials including squaring a binomial. SPI 3102.3.2 Operate with polynomials and simplify results.
a+b
b 2
a
B
b
BC
BC C
2
=
B
=
a2
+ BC +
BC
+ C2
+ ab
ab
+ b2
2
(a + b)2
+
KeyConcept Square of a Sum Words
The square of a + b is the square of a plus twice the product of a and b plus the square of b.
Symbols
(a + b) 2 = (a + b)(a + b)
Example
(x + 4) 2 = (x + 4)(x + 4)
= a 2 + 2ab + b 2
= x2 + 8x + 16
Example 1 Square of a Sum Find (3x + 5) 2. (a + b) 2 =a 2 + 2ab + b 2 (3x + 5) 2 = (3x) 2 + 2(3x)(5) + 5 2 2
= 9x + 30x + 25
Square of a sum a = 3x, b = 5 Simplify. Use FOIL to check your solution.
GuidedPractice Find each product. 1A. (8c + 3d) 2
1B. (3x + 4y) 2
connectED.mcgrawhill.com
453
There is also a pattern for the square of a difference. Write a  b as a + (b) and square it using the square of a sum pattern. (a  b) 2 = [a + (b)] 2 = a 2 + 2(a)(b) + (b) 2 = a 2  2ab + b 2
Square of a sum Simplify.
KeyConcept Square of a Difference
WatchOut! Square of a Difference Remember that (x  7) 2 does not equal x2  7 2, or x2  49. (x  7) 2 = (x  7) (x  7) = x2  14x + 49
Words
The square of a  b is the square of a minus twice the product of a and b plus the square of b.
Symbols
(a  b) 2 = (a  b)(a  b) = a 2  2ab + b 2
Example
(x  3) 2 = (x  3)(x  3) = x2  6x + 9
Example 2 Square of a Difference Find (2x  5y) 2. (a  b) 2 = a 2  2ab + b 2 (2x  5y) 2 = (2x) 2  2(2x)(5y) + (5y) 2 = 4x 2  20xy + 25y 2
Square of a difference a = 2x and b = 5y Simplify.
GuidedPractice Find each product. 2A. (6p  1) 2
2B. (a  2b) 2
The product of the square of a sum or the square of a difference is called a perfect square trinomial. We can use these to find patterns to solve realworld problems.
RealWorld Example 3 Square of a Difference PHYSICAL SCIENCE Each edge of a cube of aluminum is 4 centimeters less than each edge of a cube of copper. Write an equation to model the surface area of the aluminum cube. Let c = the length of each edge of the cube of copper. So, each edge of the cube of aluminum is c  4. SA = 6s 2 SA = 6(c  4) 2 SA = 6[c 2  2(4)(c) + 4 2] SA = 6(c 2  8c + 16)
c
c4
Formula for surface area of a cube Replace s with c  4. Square of a difference Simplify.
GuidedPractice 3. GARDENING Alano has a garden that is g feet long and g feet wide. He wants to add 3 feet to the length and the width. A. Show how the new area of the garden can be modeled by the square of a binomial. B. Find the square of this binomial.
454  Lesson 78  Special Products
2 Product of a Sum and a Difference
Now we will see what the result is when we multiply a sum and a difference, or (a + b)(a  b). Recall that a  b can be written as a + (b).
Notice that the middle terms are opposites and add to a zero pair. So (a + b)(a  b) = a 2  ab + ab  b 2 = a 2  b 2. a + (b) a a+b
zero
b 2
a
B
b
BC
BC = C
B
2
B
2
+ BC
BC
+ C 2
2
=
+ C 2
KeyConcept Product of a Sum and a Difference
StudyTip Patterns When using any of these patterns, a and b can be numbers, variables, or expressions with numbers and variables.
Words
The product of a + b and a  b is the square of a minus the square of b.
Symbols
(a + b)(a  b) = (a  b)(a + b) = a2  b2
Example 4 Product of a Sum and a Difference Find (2x 2 + 3)(2x 2  3). (a + b)(a  b) = a 2  b 2 (2x 2 + 3)(2x 2  3) = (2x 2) 2  (3) 2 = 4x 4  9
Product of a sum and difference a = 2x 2 and b = 3 Simplify.
GuidedPractice Find each product. 4A. (3n + 2)(3n  2)
Check Your Understanding
4B. (4c  7d)(4c + 7d)
= StepbyStep Solutions begin on page R12.
Examples 1–2 Find each product.
Example 3
1. (x + 5) 2
2. (11  a) 2
3 (2x + 7y) 2
4. (3m  4)(3m  4)
5. (g  4h)(g  4h)
6. (3c + 6d) 2
7. GENETICS The color of a Labrador retriever’s fur is genetic. Dark genes D are dominant over yellow genes y. A dog with genes DD or Dy will have dark fur. A dog with genes yy will have yellow fur. Pepper’s genes for fur color are Dy, and Ramiro’s are yy.
%
Z
%
%%
%Z
Z
%Z
ZZ
a. Write an expression for the possible fur colors of Pepper’s and Ramiro’s puppies. b. What is the probability that a puppy will have yellow fur? connectED.mcgrawhill.com
455
Example 4
Find each product. 8. (a  3)(a + 3)
9. (x + 5)(x  5)
10. (6y  7)(6y + 7)
11. (9t + 6)(9t  6)
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–2 Find each product. 12. (a + 10)(a + 10)
13. (b  6)(b  6)
14. (h + 7) 2
15. (x + 6) 2
16. (8  m) 2
17. (9  2y) 2
18. (2b + 3) 2
19. (5t  2) 2
20. (8h  4n) 2 Example 3
21. GENETICS The ability to roll your tongue is inherited genetically from parents if either parent has the dominant trait T. Children of two parents without the trait will not be able to roll their tongues. a. Show how the combinations can be modeled by the square of a sum.
T
t
T
TT
Tt
t
Tt
tt
b. Predict the percent of children that will have both dominant genes, one dominant gene, and both recessive genes. Example 4
Find each product. 22. (u + 3)(u  3)
23 (b + 7)(b  7)
24. (2 + x)(2  x)
25. (4  x)(4 + x)
26. (2q + 5r)(2q  5r)
27. (3a 2 + 7b)(3a 2  7b)
28. (5y + 7) 2
29. (8  10a) 2
30. (10x  2)(10x + 2)
4b) 2
31. (3t + 12)(3t  12)
32. (a +
34. (2c  9d) 2
35. (g + 5h) 2
37. (3a  b)(3a + b) 4
4
3 40. _ k+8
)
(4
2
36. (6y  13)(6y + 13)
38. (5x  y
2 2
39. (8a 2  9b 3)(8a 2 + 9b 3)
2 41. _ y4
2
42. (7z 2 + 5y 2)(7z 2  5y 2)
2
(5
43. (2m + 3)(2m  3)(m + 4)
33. (3q  5r) 2
)
)
44. (r + 2)(r  5)(r  2)(r + 5)
45. GEOMETRY Write a polynomial that represents the area of the figure at the right.
B
46. FLYING DISKS A flying disk shaped like a circle has a radius of x + 3 inches. a. Write an expression representing the area of the flying disk. b. If the diameter of the flying disk is 8 inches, what is its area? GEOMETRY Find the area of each shaded region. 47.
48. x1
x+6 x+2
x1 x3 x+2 x3
456  Lesson 78  Special Products
x+5
x1 x1
x+2 x+2
Find each product. 49. (c + d)(c + d)(c + d)
50. (2a  b) 3
51. (f + g)(f  g)(f + g)
52. (k  m)(k + m)(k  m)
53. (n  p) 2(n + p)
54. (q + r) 2(q  r)
55 WRESTLING A high school wrestling mat must be a square with 38foot sides and contain two circles as shown. Suppose the inner circle has a radius of r feet, and the radius of the outer circle is nine feet longer than the inner circle.
38 ft
a. Write an expression for the area of the larger circle. b. Write an expression for the area of the portion of the square outside the larger circle.
C
56.
a
MULTIPLE REPRESENTATIONS In this problem, you will investigate a pattern. Begin with a square piece of construction paper. Label each edge of the paper a. In any of the corners, draw a smaller square and label the edges b.
a b
a
b
a
b
b
a. Numerical Find the area of each of the squares.
a
b. Concrete Cut the smaller square out of the corner. What is the area of the shape?
ab b
c. Analytical Remove the smaller rectangle on the bottom. Turn it and slide it next to the top rectangle. What is the length of the new arrangement? What is the width? What is the area?
ab ?
d. Analytical What pattern does this verify? ?
H.O.T. Problems
Use HigherOrder Thinking Skills
57. WHICH ONE DOESN’T BELONG? Which expression does not belong? Explain.
(2c  d )(2c  d )
(2c + d )(2c  d )
(2c + d )(2c + d )
(c + d )(c + d )
58. CHALLENGE Does a pattern exist for the cube of the sum (a + b) 3? a. Investigate this question by finding the product (a + b)(a + b)(a + b). b. Use the pattern you discovered in part a to find (x + 2) 3. c. Draw a diagram of a geometric model for the cube of a sum. d. What is the pattern for the cube of a difference, (a  b) 3? 59. REASONING Find c that makes 25x 2  90x + c a perfect square trinomial. 60. OPEN ENDED Write two binomials with a product that is a binomial. Then write two binomials with a product that is not a binomial. 61. WRITING IN MATH Describe how to square the sum of two quantities, square the difference of two quantities, and how to find the product of a sum of two quantities and a difference of two quantities. connectED.mcgrawhill.com
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SPI 3108.4.3, SPI 3102.3.2, SPI 3102.1.2
Standardized Test Practice −− 62. GRIDDED RESPONSE In the right triangle, DB bisects ∠B. What is the measure of ∠ADB in degrees?
64. Myron can drive 4 miles in m minutes. At this rate, how many minutes will it take him to drive 19 miles?
"
4m H _
F 76m 50°
19
19m G _ 4
%
65. What property is illustrated by the equation 2x + 0 = 2x?
$
#
A B C D
63. What is the product of (2a  3) and (2a  3)? A 4a 2 + 12a + 9 B 4a 2 + 9
4 J _ 19m
C 4a 2  12a  9 D 4a 2  12a + 9
Commutative Property of Addition Additive Inverse Property Additive Identity Property Associative Property of Addition
Spiral Review Find each product. (Lesson 77) 66. (y  4)(y  2)
67. (2c  1)(c + 3)
68. (d  9)(d + 5)
69. (4h  3)(2h  7)
70. (3x + 5)(2x + 3)
71. (5m + 4)(8m + 3)
72. x(2x  7) + 5x
73. c(c  8) + 2c(c + 3)
74. 8y(3y + 7)  11y 2
75. 2d(5d)  3d(d + 6)
76. 5m(2m 3 + m 2 + 8) + 4m
1 2 77. 3p(6p  4) + 2 _ p  3p
Simplify. (Lesson 76)
(2
)
Use substitution to solve each system of equations. (Lesson 62) 78. 4c = 3d + 3 c=d1
79. c  5d = 2 2c + d = 4
80. 5r  t = 5 4r + 5t = 17
81. BIOLOGY Each type of fish thrives in a specific range of temperatures. The best temperatures for sharks range from 18°C to 22°C, inclusive. Write a compound inequality to represent temperatures where sharks will not thrive. (Lesson 62) Write an equation of the line that passes through each pair of points. (Lesson 42) 82. (1, 1), (7, 4)
83. (5, 7), (0, 6)
$6.40 .440 40/ 0/lbb 0/l
84. (5, 1), (8, 2)
85. COFFEE A coffee store wants to create a mix using two coffees. How many pounds of coffee A should be mixed with 9 pounds of coffee B to get a mixture that can sell for $6.95 per pound? (Lesson 29)
Skills Review Find the prime factorization of each number. (Concepts and Skills Bank Lesson 3) 86. 40
87. 120
458  Lesson 78  Special Products
88. 900
89. 165
$7.28//lbb
Study Guide and Review Study Guide KeyConcepts
KeyVocabulary
For any nonzero real numbers a and b and any integers m, n, and p, the following are true.
binomial (p. 424)
Multiplying Monomials (Lesson 71) • Product of Powers: a m · a n = a m + n • Power of a Power: (a m) n = a m · n • Power of a Product: (ab) m = a mb m
degree of a monomial (p. 424)
constant (p. 401)
degree of a polynomial (p. 424) FOIL method (p. 448) leading coefficient (p. 425)
Dividing Monomials (Lesson 72) am mp • Quotient of Powers: _ p =a
monomial (p. 401) order of magnitude (p. 411)
a
m am • Power of a Quotient: _a = _ b bm
()
polynomial (p. 424)
• Zero Exponent: a 0 = 1 1 1 n _ • Negative Exponent: a n = _ n and n = a
quadratic expression (p. 448)
Scientific Notation (Lesson 73) • A number is in scientific notation if it is in the form a × 10 n, where 1 ≤ a < 10. • To write in standard form: • If n > 0, move the decimal n places right. • If n < 0, move the decimal n places left.
standard form of a polynomial (p. 425)
a
scientific notation (p. 416)
a
Operations with Polynomials (Lessons 75 through 78) • To add or subtract polynomials, add or subtract like terms. To multiply polynomials, use the Distributive Property. • Special products: (a + b) 2 = a 2 + 2ab + b 2 (a  b) 2 = a 2  2ab + b 2 (a + b)(a  b) = a 2  b 2
VocabularyCheck Choose a term from the Key Vocabulary list above that best describes each expression or equation. 1. x 2 + 1 2. 5 0 = 1 3. x 2  3x + 2
StudyOrganizer
4. (xy 3)(x 2y 4) = x 3y 7
Be sure the Key Concepts are noted in your Foldable. Poly nom
trinomial (p. 424)
71 72 73 74
ials
3 5. (a 7) = a 21
1 6. 5 2 = _ 2 5
7. 6.2 × 10 5 8. (x + 2)(x  5) = x 2  3x  10 9. x 3 + 2x 2  3x  1 10. 7xy 4 connectED.mcgrawhill.com
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Study Guide and Review Continued LessonbyLesson Review
711 Multiplying Monomials
(pp. 401–407)
Example 1
Simplify each expression. 11. x · x 3 · x 5
12. (2xy)(3x 2y 5)
13. (4ab 4)(5a 5b 2)
14. (6x 3y 2)
3 2
✔3102.3.4, CLE 3102.4.1
2
15. ⎡⎣(2r 3t) ⎤⎦
16. (2u 3)(5u)
3 3 17. (2x 2) (x 3)
1 ( 3)3 18. _ 2x 2
19. GEOMETRY Use the formula V = πr 2h to find the volume of the cylinder.
Simplify (5x 2y 3)(2x 4y).
(5x 2y 3)(2x 4y) = (5 · 2)(x 2 · x 4)(y 3 · y)
Commutative Property
= 10x 6y 4
Product of Powers
Example 2 3
Simplify (3a 2b 4) .
(3a 2b 4)3 = 3 3(a 2)3(b 4)3
3x
= 27a 6b 12
Power of a Product Simplify.
2
5x
✔3102.3.4
722 Dividing Monomials
(pp. 408–415)
Simplify each expression. Assume that no denominator equals zero. 3 3
(3x)0
20. _
21.
2a
4
12y 22. _ 5
23. a 3b 0c 6
3y
1)2
(3x 25. _ (3x 2)2
7 8 4
15x y z 24. _ 3 5 3 45x y z
26.
(
11 9
6xy z _ 48x 6yz 7
3xy ) (_ 2z
)
0
( 2 )( y )( x )
y4 12 _ x _ 27. _ 5 4
28. GEOMETRY The area of a rectangle is 25x 2y 4 square feet. The width of the rectangle is 5xy feet. What is the length of the rectangle? 5xy
Example 3
_
4 3 Simplify 2k m . Assume that no denominator equals zero. 2
4k m k4 _ m3 _ = _2 _ m 2 2 4 k 4k m
2k 4m 3
( )( )( ) 1 = (_ k m 2) 42
2m 2 = k_
Quotient of Powers Simplify.
2
Example 4
_
4 2 uv Simplify t 3 . Assume that no denominator equals zero. 7
t u 4 2 u ( 2) t_ uv t4 _ = _ v 3 u 7 3 7 t t u
( )( )
= (t 4 + 3)(u 1  7)(v 2) 7 6 2
=t u
t7 = _ 6 2 u v
460  Chapter 7  Study Guide and Review
31
Group powers with the same base.
v
Group the powers with the same base. Quotient of Powers Simplify. Simplify.
✔3102.2.5, ✔3102.2.6, SPI 3102.2.2
733 Scientific Notation
(pp. 416–422)
Express each number in scientific notation.
Example 5
29. 2,300,000
Express 300,000,000 in scientific notation.
30. 0.0000543
31. ASTRONOMY Earth has a diameter of about 8000 miles. Jupiter has a diameter of about 88,000 miles. Write in scientific notation the ratio of Earth’s diameter to Jupiter’s diameter.
Step 1 300,000,000
3.00000000 Step 2 The decimal point moved 8 places to the left, so n = 8. Step 3 300,000,000 = 3 × 10 8
CLE 3102.3.2, SPI 3102.3.2
744 Polynomials
(pp. 424–429)
Write each polynomial in standard form.
Example 6
32. x + 2 + 3x 2
33. 1  x 4
Write 3  x 2 + 4x in standard form.
34. 2 + 3x + x 2
35. 3x5  2 + 6x  2x2 + x3
36. GEOMETRY Write a polynomial that represents the perimeter of the figure. 3x
3 2
x 1
4x 5x
2
4x
1
x
Step 1 Find the degree of each term.
3: degree 0 2 degree 2 x : 4x: degree 1 Step 2 Write the terms in descending order of degree. 3  x 2 + 4x = x 2 + 4x + 3
6x 8x
4
CLE 3102.3.2, ✔3102.3.5, SPI 3102.3.2
755 Adding and Subtracting Polynomials
(pp. 433–438)
Find each sum or difference.
Example 7
37. (x 3 + 2) + (3x 3  5)
Find (8r 2 + 3r)  (10r 2  5).
38. a 2 + 5a  3  (2a 2  4a + 3)
(8r 2 + 3r)  (10r 2  5) = (8r 2 + 3r) + (10r 2 + 5) = (8r 2  10r 2 ) + 3r + 5
39. (4x  3x 2 + 5) + (2x 2  5x + 1) 40. (6ab + 3b 2)  (3ab  2b 2)
2
= 2r + 3r + 5
41. PICTURE FRAMES Jean is framing a painting that is a rectangle. What is the perimeter of the frame?
Use the additive inverse. Group like terms. Add like terms.
5x + 3
2
2x  3x + 1
connectED.mcgrawhill.com
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Study Guide and Review Continued CLE 3102.3.2, SPI 3102.3.2
766 Multiplying a Polynomial by a Monomial
(pp. 439–444)
Solve each equation.
Example 8
42. x 2(x + 2) = x(x 2 + 2x + 1)
Solve m (2m  5) + m = 2m (m  6) + 16.
43. 2x (x + 3) = 2(x 2 + 3)
m (2m  5) + m = 2m (m  6) + 16
44. 2(4w + w 2)  6 = 2w (w  4) + 10
2m 2  5m + m = 2m 2  12m + 16 2m 2  4m = 2m 2  12m + 16
45. 6k (k + 2) = 6(k 2 + 4)
4m = 12m + 16
46. GEOMETRY Find the area of the rectangle.
3x
8m = 16 m=2
2
x +x7
CLE 3102.3.2, ✔3102.3.5, SPI 3102.3.2
777 Multiplying Polynomials
(pp. 447–452)
Example 9
Find each product. 47. (x  3)(x + 7)
48. (3a  2)(6a + 5)
Find (6x  5)(x + 4).
49. (3r  7t )(2r + 5t )
50. (2x + 5)(5x + 2)
(6x  5)(x + 4)
51. PARKING LOT The parking lot shown is to be paved. What is the area to be paved?
2x + 3
F
O
I
L
= (6x )(x ) + (6x )(4) + (5)(x ) + (5)(4) = 6x 2 + 24x  5x  20
Multiply.
= 6x 2 + 19x  20
Combine like terms.
5x  4 CLE 3102.3.2, ✔3102.3.5, SPI 3102.3.2
788 Special Products
(pp. 453–458)
Example 10
Find each product. 52. (x + 5)(x  5)
53. (3x  2) 2
Find (x  7) 2.
54. (5x + 4) 2
55. (2x  3)(2x + 3)
56. (2r + 5t ) 2
57. (3m  2)(3m + 2)
(a  b) 2 = a 2  2ab + b 2 (x  7) 2 = x 2  2(x )(7) + (7) 2 = x 2  14x + 49
58. GEOMETRY Write an expression to represent the area of the shaded region.
Simplify.
Example 11 (a + b)(a  b) = a 2  b 2
x 2
a = x and b = 7
Find (5a  4)(5a + 4).
2x + 5 x +2
Square of a Difference
2x  5
462  Chapter 7  Study Guide and Review
(5a  4)(5a + 4) = (5a) 2  (4) 2 = 25a 2  16
Product of a Sum and Difference a = 5a and b = 4 Simplify.
Tennessee Curriculum Standards
Practice Test Simplify each expression.
SPI 3102.1.3, SPI 3102.3.2
Find each sum or difference.
1. (x 2)(7x 8)
14. (x + 5) + (x 2  3x + 7)
2. (5a 7bc 2)(6a 2bc 5)
15. (7m  8n 2 + 3n)  (2n 2 + 4m  3n)
3. MULTIPLE CHOICE Express the volume of the solid as a monomial.
16. MULTIPLE CHOICE Antonia is carpeting two of the rooms in her house. The dimensions are shown. What is the total area to be carpeted? x
x 2
x x x
A x3
C 6x 3
B 6x
D x6
Simplify each expression. Assume that no denominator equals 0.
F x 2 + 3x
H x 2 + 3x  5
G 2x 2 + 6x  10
J 8x + 12
Find each product. 17. a(a 2 + 2a  10) 18. (2a  5)(3a + 5)
6 8
19. (x  3)(x 2 + 5x  6)
x
20. (x + 3) 2
x y 4. _ 2 2a 4b 3 0
( )
5. _ 6 c
7
2xy 6. _ 8x
Express each number in scientific notation. (Lesson 73)
7. 0.00021 8. 58,000 Express each number in standard form. 9. 2.9 × 10 5 10. 9.1 × 10 6 Evaluate each product or quotient. Express the results in scientific notation. 11. (2.5 × 10 3)(3 × 10 4) 8.8 × 10 2 12. _ 4 4 × 10
13. ASTRONOMY The average distance from Mercury to the Sun is 35,980,000 miles. Express this distance in scientific notation.
x +5
x +3
21. (2b  5)(2b + 5) 22. GEOMETRY A rectangular prism has dimensions x, x + 3, and 2x + 5. a. Find the volume of the prism in terms of x. b. Choose two values for x. How do the volumes compare? Solve each equation. 23. 5(t 2  3t + 2) = t(5t  2) 24. 3x(x + 2) = 3(x 2  2) 25. FINANCIAL LITERACY Money invested in a certificate of deposit (CD) earns interest once per year. Suppose you invest $4000 in a 2year CD. a. If the interest rate is 5% per year, the expression 4000(1 + 0.05) 2 can be evaluated to find the total amount of money after two years. Explain the numbers in this expression. b. Find the amount at the end of two years. c. Suppose you invest $10,000 in a CD for 4 years at an annual rate of 6.25%. What is the total amount of money you will have after 4 years? connectED.mcgrawhill.com
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Preparing for Standardized Tests Using a Scientific Calculator Scientific calculators are powerful problemsolving tools. There are times when using a scientific calculator can be used to make computations faster and easier, such as computations with very large numbers. However, there are times when using a scientific calculator is necessary, like the estimation of irrational numbers.
Strategies for Using a Scientific Calculator Step 1
Familiarize yourself with the various functions of a scientific calculator as well as when they should be used: • Exponents scientific notation, calculating with large or small numbers • Pi solving circle problems, like circumference and area • Square roots distance on a coordinate plane, Pythagorean theorem • Graphs analyzing paired data in a scatter plot, graphing functions, finding roots of equations
Step 2
Use your scientific or graphing calculator to solve the problem. • Remember to work as efficiently as possible. Some steps may be done mentally or by hand, while others should be completed using your calculator. • If time permits, check your answer.
SPI 3102.2.2
Test Practice Example Read the problem. Identify what you need to know. Then use the information in the problem to solve. The distance from the Sun to Jupiter is approximately 7.786 × 10 11 meters. If the speed of light is about 3 × 10 8 meters per second, how long does it take for light from the Sun to reach Jupiter? Round to the nearest minute. A about 43 minutes
C about 1876 minutes
B about 51 minutes
D about 2595 minutes
464  Chapter 7  Preparing for Standardized Tests
Read the problem carefully. You are given the approximate distance from the Sun to Jupiter as well as the speed of light. Both quantities are given in scientific notation. You are asked to find how many minutes it takes for light from the Sun to reach Jupiter. Use the relationship distance = rate × time to find the amount of time. d=r×t
_d = t r
To find the amount of time, divide the distance by the rate. Notice, however, that the units for time will be seconds. 7.786 × 10 11 m __ = t seconds 3 × 10 8 m/s
Use a scientific calculator to quickly find the quotient. On most scientific calculators, the EE key is used to enter numbers in scientific notation. KEYSTROKES:
7.786
11
3
8
The result is 2595.33333333 seconds. To convert this number to minutes, use your calculator to divide the result by 60. This gives an answer of about 43.2555 minutes. The answer is A.
Exercises Read each problem. Identify what you need to know. Then use the information in the problem to solve. 1. Since its creation 5 years ago, approximately 2.504 × 10 7 items have been sold or traded on a popular online website. What is the average daily number of items sold or traded over the 5year period? A about 9640 items per day
3. The population of the United States is about 3.034 × 10 8 people. The land area of the country is about 3.54 × 10 6 square miles. What is the average population density (number of people per square mile) of the United States? A about 136.3 people per square mile B about 112.5 people per square mile C about 94.3 people per square mile D about 85.7 people per square mile
B about 13,720 items per day C about 1,025,000 items per day D about 5,008,000 items per day 2. Evaluate √ ab if a = 121 and b = 23.
4. Eleece is making a cover for the marching band’s bass drum. The drum has a diameter of 20 inches. Estimate the area of the face of the bass drum.
F about 5.26
F 31.41 square inches
G about 9.90
G 62.83 square inches
H about 12
H 78.54 square inches
J about 52.75
J 314.16 square inches connectED.mcgrawhill.com
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Standardized Test Practice Cumulative, Chapters 1 through 7 4. Express the perimeter of the rectangle below as a polynomial.
Multiple Choice Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
2
x  3x + 4
1. Express the area of the triangle below as a monomial.
2
2x  x + 3
F 3x 2  4x + 7 G 3x 2 + x + 7
2 5
3b d
H 6x 2  8x + 14 3 2
J 6x 2  4x + 7
4b d
A 12b 5d 7 B 12b 6d 10 C 6b 6d 10 D
5. Subtract the polynomials below.
6b 5d 7
(7a 2 + 6a  2)  (4a 3 + 3a 2 + 5) A 4a 3 + 4a 2 + 6a  7
2. Simplify the following expression. 2 5 3
( _) 2w z 3y 4
B 11a 2 + 3a  7 C 4a 3 + 10a 2 + 6a + 3 D 4a 3 + 7a 3  3a
2w 5z 8 F _ 7 3y
8w 6z 15 G _ 12 27y
6. Which inequality is shown in the graph?
8w 5z 8 H _ 7 27y
y
2w 6z 15 J _ 12 3y
0
3. Which equation of a line is perpendicular to 3 x  3? y=_ 5
5 A y = _ x+2 3 _ B y = 3x + 2 5
5 C y=_ x2 3
3 D y=_ x2 5
2 F y ≤ _ x1 3
3 G y ≤ _ x1 4
TestTakingTip Question 2 Use the laws of exponents to simplify the expression. Remember, to find the power of a power, multiply the exponents.
2 x+1 H y ≤ _ 3
3 J y ≤ _ x+1 4
466  Chapter 7  Standardized Test Practice
x
Short Response/Gridded Response 7. Mickey has 180 feet of fencing that she wants to use to enclose a play area for her puppy. She will use her house as one of the sides of the region.
10. GRIDDED RESPONSE At a family fun center, the Wilson and Sanchez families each bought video game tokens and batting cage tokens as shown in the table. Family
180  2x ft
x ft
Wilson
Sanchez
Number of Video Game Tokens
25
30
Number of Batting Cage Tokens
8
6
$26.50
$25.50
Total Cost
x ft
What is the cost in dollars of a batting cage token at the family fun center?
House
a. If she makes the play area x feet deep as shown in the figure, write a polynomial in standard form to represent the area of the region. b. How many square feet of area will the puppy have to play in if Mickey makes it 40 feet deep?
Extended Response Record your answers on a sheet of paper. Show your work.
8. Identify the expression below that does not belong with the other two. Explain.
11. The table below shows the distances from the Sun to Mercury, Earth, Mars, and Saturn. Use the data to answer each question.
(3m  2n )(3m + 2n )
Planet
(3m + 2n )(3m + 2n )
(3m + 2n )(3m  2n ) 9. What is the solution to the following system of equations? Show your work.
Distance from Sun (km)
Mercury
5.79 × 10 7
Earth
1.50 × 10 8
Mars
2.28 × 10 8
Saturn
1.43 × 10 9
a. Of the planets listed, which one is the closest to the Sun?
⎧y = 6x  1 ⎨y = 6x + 4 ⎩
b. About how many times as far from the Sun is Mars as Earth?
Need ExtraHelp? If you missed Question... Go to Lesson... For help with TN SPI...
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