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3. Lesson 8-1 Multiplying Monomials Lesson 8-2 Dividing Monomials
Lesson 8-3 Scientific Notation
Lesson 8-4 Polynomials
Lesson 8-5 Adding and Subtracting Polynomials
Lesson 8-6 Multiplying Polynomials by a Monomial
Lesson 8-7 Multiplying Polynomials
Lesson 8-8 Special Products
Contents

4. Example 1 Identify Monomials Example 2 Product of Powers
Example 3 Power of a Power
Example 4 Power of a Product
Example 5 Simplify Expressions
Lesson 1 Contents

5. Determine whether each expression is a monomial. Explain your reasoning.xy d. c. b. a.
Reason
Monomial?
Expression no
The expression involves subtraction, not the product, of two variables.
yes
The expression is the product of a number and two variables.
yes
is a real number and an example of a constant.
yes
The expression is the product of two variables.
Example 1-1a

6. Determine whether each expression is a monomial. Explain your reasoning.b. a.
Reason
Monomial?
Expression
yes
Single variables are monomials. no
The expression involves subtraction, not the product, of two variables. no
The expression is the quotient, not the product, of two variables.
yes
The expression is the product of a number, , and two variables.
Example 1-1b

7. Commutative and Associative Properties
Simplify .
Commutative and Associative Properties
Product of Powers
Simplify.
Answer:
Example 1-2a

8. Commutative and Associative Properties
Simplify .
Commutative and Associative Properties
Product of Powers
Simplify.
Answer:
Example 1-2b

9. Simplify each expression. a.b.
Answer:
Answer:
Example 1-2c

10. Simplify Power of a Power Simplify. Power of a Power Simplify. Answer:
Example 1-3a

11. Simplify
Answer:
Example 1-3b

12. Geometry Find the volume of a cube with a side length
Formula for volume of a cube
Power of a Product
Simplify.
Answer:
Example 1-4a

13. Express the surface area of the cube as a monomial.
Answer:
Example 1-4b

14. Simplify Power of a Power Power of a Product Power of a Power
Example 1-5a

15. Commutative Property
Answer:
Power of Powers
Example 1-5b

16. Simplify
Answer:
Example 1-5c

17. End of Lesson 1

18. Example 1 Quotient of Powers Example 2 Power of a Quotient
Example 3 Zero Exponent
Example 4 Negative Exponents
Example 5 Apply Properties of Exponents
Lesson 2 Contents

19. Simplify Assume that x and y are not equal to zero.
Group powers that have the same base.
Quotient of Powers
Answer:
Simplify.
Example 2-1a

20. Simplify Assume that a and b are not equal to zero.
Answer:
Example 2-1b

21. Simplify Assume that e and f are not equal to zero.
Power of a Quotient
Power of a Product
Power of a Power
Answer:
Example 2-2a

22. Simplify Assume that p and q are not equal to zero.
Answer:
Example 2-2b

23. Simplify Assume that m and n are not equal to zero.
Answer: 1
Example 2-3a

24. Simplify . Assume that m and n are not equal to zero.
Answer:
Quotient of Powers
Example 2-3b

25. Simplify each expression. Assume that z is not equal to zero.b.
Answer: 1
Answer:
Example 2-3c

26. Simplify . Assume that x, y, and z are not equal to zero.
Write as a product of fractions.
Answer:
Multiply fractions.
Example 2-4a

27. Simplify . Assume that p, q, and r are not equal to zero.
Group powers with the same base.
Quotient of Powers and Negative Exponent Properties
Example 2-4b

28. Negative Exponent Property
Simplify.
Negative Exponent Property
Multiply fractions.
Answer:
Example 2-4c

29. Simplify each expression. Assume that no denominator is equal to zero.b.
Answer:
Answer:
Example 2-4d

30. Multiple-Choice Test Item
Write the ratio of the circumference of the circle to the area of the square in simplest form.
A B C D
Read the Test Item A ratio is a comparison of two quantities. It can be written in fraction form.
Example 2-5a

31. Solve the Test Item
circumference of a circle side length of the square diameter of circle or 2r area of square
Substitute.
Quotient of Powers
Example 2-5b

32. Simplify.
Answer: C
Example 2-5c

33. Multiple-Choice Test Item
Write the ratio of the circumference of the circle to the perimeter of the square in simplest form.
A B C D
Answer: A
Example 2-5d

34. End of Lesson 2

35. Example 1 Scientific to Standard Notation
Example 2 Standard to Scientific Notation
Example 3 Use Scientific Notation
Example 4 Multiplication with Scientific Notation
Example 5 Division with Scientific Notation
Lesson 3 Contents

36. Express in standard notation.
move decimal point 3 places to the left.
Answer:
Example 3-1a

37. Express in standard notation.
move decimal point 5 places to the right.
Answer: 219,000
Example 3-1b

38. Express each number in standard notation. a.
Answer:
Answer: 7610
Example 3-1c

39. Express 0.000000672 in scientific notation.
Move decimal point 7 places to the right.
and
Answer:
Example 3-2a

40. Express 3,022,000,000,000 in scientific notation.
Move decimal point 12 places to the left.
and
Answer:
Example 3-2b

41. Express each number in scientific notation. a. 458,000,000
Answer:
Answer:
Example 3-2c

42. Answer: Shoes sold to women:
The Sporting Goods Manufacturers Association reported that in 2000, women spent $4.4 billion on 124 million pairs of shoes. Men spent $8.3 billion on 169 million pairs of shoes.
Express the numbers of pairs of shoes sold to women, pairs sold to men, and total spent by both men and women in standard notation.
Answer: Shoes sold to women:
Shoes sold to men:
Total spent:
Example 3-3a

43. Write each of these numbers in scientific notation.
Answer: Shoes sold to women:
Shoes sold to men:
Total spent:
Example 3-3b

44. The average circulation for all U. S. daily newspapers in 2000 was 111
The average circulation for all U.S. daily newspapers in 2000 was billion newspapers. The top three leading newspapers were The Wall Street Journal, with a circulation of 1.76 million newspapers, USA Today, which sold 1.69 million newspapers, and The New York Times, which had 1.10 million readers.
a. Express the average daily circulation and the circulation of the top three newspapers in standard notation.
Answer: Total circulation: 111,500,000,000; The Wall Street Journal: 1,760,000; USA Today: 1,690,000; The New York Times: 1,100,000
Example 3-3c

45. b. Write each of the numbers in scientific notation.
The average circulation for all U.S. daily newspapers in 2000 was billion newspapers. The top three leading newspapers were The Wall Street Journal, with a circulation of 1.76 million newspapers, USA Today, which sold 1.69 million newspapers, and The New York Times, which had 1.10 million readers.
b. Write each of the numbers in scientific notation.
Answer: Total circulation: The Wall Street Journal: 1.76 USA Today: The New York Times:
Example 3-3d

46. Evaluate Express the result in scientific and standard notation.
Commutative and Associative Properties
Product of Powers
Associative Property
Example 3-4a

47. Product of Powers
Answer:
Example 3-4b

48. Evaluate Express the result in scientific and standard notation.
Answer:
Example 3-4c

49. Evaluate Express the result in scientific and standard notation.
Associative Property
Product of Powers
Answer:
Example 3-5a

50. Evaluate Express the result in scientific and standard notation.
Answer:
Example 3-5b

51. End of Lesson 3

52. Example 1 Identify Polynomials Example 2 Write a Polynomial
Example 3 Degree of a Polynomial
Example 4 Arrange Polynomials in Ascending Order
Example 5 Arrange Polynomials in Descending Order
Lesson 4 Contents

53. Monomial, Binomial, or Trinomial
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
Monomial, Binomial, or Trinomial
Polynomial?
Expression a. b. c. d.
Yes, is the difference of two real numbers.
binomial
Yes, is the sum and difference of three monomials.
trinomial
No are not monomials.
none of these
Yes, has one term.
monomial
Example 4-1a

54. Monomial, Binomial, or Trinomial
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
Monomial, Binomial, or Trinomial
Polynomial?
Expression a. b. c. d.
Yes, is the sum of three monomials.
trinomial
No which is not a monomial.
none of these
Yes, The expression is the sum of two monomials.
binomial
Yes, has one term.
monomial
Example 4-1b

55. Write a polynomial to represent the area of the green shaded region.
Words The area of the shaded region is the area of the rectangle minus the area of the triangle.
Variables area of the shaded region height of rectangle area of rectangle triangle area
Example 4-2a

56. Equation A A
Answer: The polynomial representing the area of the shaded region is
Example 4-2b

57. Write a polynomial to represent the area of the green shaded region.
Answer:
Example 4-2c

58. Find the degree of each polynomial.b. a.
Degree of Polynomial
Degree of Each Term
Terms
Polynomial
0, 1, 2, 3 3
2, 1, 0 2 8 8
Example 4-3a

59. Find the degree of each polynomial.b. a.
Degree of Polynomial
Degree of Each Term
Terms
Polynomial
2ac2, –7
2, 1, 3, 0 3
2, 4, 3 4 –
7, 6 7 –
Example 4-3b

60. Arrange the terms of so that the powers of x are in ascending order.
Answer:
Example 4-4a

61. Arrange the terms of so that the powers of x are in ascending order.
Answer:
Example 4-4b

62. Arrange the terms of each polynomial so that the powers of x are in ascending order.b.
Answer: 1 – 2x + 6x2 – 3x4
Answer:
Example 4-4c

63. Arrange the terms of so that the powers of x are in descending order.
Answer:
Example 4-5a

64. Arrange the terms of so that the powers of x are in descending order.
Answer:
Example 4-5b

65. Arrange the terms of each polynomial so that the powers of x are in descending order.b.
Answer:
Answer:
Example 4-5c

66. End of Lesson 4

67. Example 1 Add Polynomials Example 2 Subtract Polynomials
Lesson 5 Contents

68. Group like terms together.
Find
Method 1 Horizontal
Group like terms together.
Associative and Commutative Properties
Add like terms.
Example 5-1a

69. Align the like terms in columns and add.
Method 2 Vertical
Align the like terms in columns and add.
Notice that terms are in descending order with like terms aligned.
Answer:
Example 5-1b

70. Find
Answer:
Example 5-1c

71. Subtract by adding its additive inverse.
Find
Method 1 Horizontal
Subtract by adding its additive inverse.
The additive inverse of is
Group like terms.
Add like terms.
Example 5-2a

72. Method 2 Vertical
Align like terms in columns and subtract by adding the additive inverse.
Add the opposite.
Answer: or
Example 5-2b

73. Find
Answer:
Example 5-2c

74. Geometry The measure of the perimeter of the triangle shown is
Find the polynomial that represents the third side of the triangle.
Let a = length of side 1, b = the length of side 2, and c = the length of the third side.
You can find a polynomial for the third side by subtracting side a and side b from the polynomial for the perimeter.
Example 5-3a

75. To subtract, add the additive inverses.
Example 5-3b

76. Answer: The polynomial for the third side is
Group the like terms.
Add like terms.
Answer: The polynomial for the third side is
Example 5-3c

77. Find the length of the third side if the triangle if
The length of the third side is
Simplify.
Answer: 45 units
Example 5-3d

78. a. Find a polynomial that represents the width of the rectangle.
Geometry The measure of the perimeter of the rectangle shown is 10r – 60.
a. Find a polynomial that represents the width of the rectangle.
b. Find the width of the rectangle if
Answer:
Answer: 3 units
Example 5-3e

79. End of Lesson 5

80. Example 1 Multiply a Polynomial by a Monomial
Example 2 Simplify Expressions
Example 3 Use Polynomial Models
Example 4 Polynomials on Both Sides
Lesson 6 Contents

81. Distributive Property
Find
Method 1 Horizontal
Distributive Property
Multiply.
Example 6-1a

82. Distributive Property
Find
Method 2 Vertical
Distributive Property
Multiply.
Answer:
Example 6-1b

83. Find
Answer:
Example 6-1c

84. Distributive Property
Simplify
Distributive Property
Product of Powers
Commutative and Associative Properties
Combine like terms.
Answer:
Example 6-2a

85. Simplify
Answer:
Example 6-2b

86. Find an expression for how much money Sarita spent at the park.
Entertainment Admission to the Super Fun Amusement Park is $10. Once in the park, super rides are an additional $3 each and regular rides are an additional $2. Sarita goes to the park and rides 15 rides, of which s of those 15 are super rides.
Find an expression for how much money Sarita spent at the park.
Words The total cost is the sum of the admission, super ride costs, and regular ride costs.
Variables If the number of super rides, then is the number of regular rides. Let M be the amount of money Sarita spent at the park.
Example 6-3a

87. M 10 s 3 2 Equation Distributive Property Simplify. Simplify.
Amount of money
equals
admission
plus
super rides
times
$3 per ride
regular rides
$2 per ride. M 10 s 3 2
Distributive Property
Simplify.
Simplify.
Answer: An expression for the amount of money Sarita spent in the park is , where s is the number of super rides she rode.
Example 6-3b

88. Evaluate the expression to find the cost if Sarita rode 9 super rides.
Add.
Answer: Sarita spent $49.
Example 6-3c

89. a. Find an expression for how much rent the Fosters received.
The Fosters own a vacation home that they rent throughout the year. The rental rate during peak season is $120 per day and the rate during the off-peak season is $70 per day. Last year they rented the house 210 days, p of which were during peak season.
a. Find an expression for how much rent the Fosters received.
b. Evaluate the expression if p is equal to 130.
Answer:
Answer: $21,200
Example 6-3d

90. Distributive Property
Solve
Original equation
Distributive Property
Combine like terms.
Subtract from each side.
Example 6-4a

91. Add 7 to each side. Add 2b to each side. Divide each side by 14.
Answer:
Example 6-4b

92. Check Original equation Simplify. Multiply. Add and subtract.
Example 6-4c

93. Solve
Answer:
Example 6-4d

94. End of Lesson 6

95. Example 1 The Distributive Property Example 2 FOIL Method
Lesson 7 Contents

96. Find
Method 1 Vertical
Multiply by –4.
Example 7-1a

97. Find
Multiply by y.
Example 7-1b

98. Find
Add like terms.
Example 7-1c

99. Distributive Property
Find
Method 2 Horizontal
Distributive Property
Distributive Property
Multiply.
Combine like terms.
Answer:
Example 7-1d

100. Find
Answer:
Example 7-1e

101. Find F L O I
Multiply.
Combine like terms.
Answer:
Example 7-2a

102. Find F I O L
Multiply.
Answer:
Combine like terms.
Example 7-2b

103. Find each product. a. b.
Answer:
Answer:
Example 7-2c

104. Geometry The area A of a triangle is one-half the height h times the base b. Write an expression for the area of the triangle.
Identify the height and the base.
Now write and apply the formula.
Area equals one-half height times base. A h b
Example 7-3a

105. Original formula
Substitution
FOIL method
Multiply.
Example 7-3b

106. Distributive Property
Combine like terms.
Distributive Property
Answer: The area of the triangle is square units.
Example 7-3c

107. Geometry The area of a rectangle is the measure of the base times the height. Write an expression for the area of the rectangle.
Answer:
Example 7-3d

108. Distributive Property
Find
Distributive Property
Distributive Property
Answer:
Combine like terms.
Example 7-4a

109. Distributive Property
Find
Distributive Property
Distributive Property
Answer:
Combine like terms.
Example 7-4b

110. Find each product. a. b.
Answer:
Answer:
Example 7-4c

111. End of Lesson 7

112. Example 2 Square of a Difference Example 3 Apply the Sum of a Square
Example 1 Square of a Sum
Example 2 Square of a Difference
Example 3 Apply the Sum of a Square
Example 4 Product of a Sum and a Difference
Lesson 8 Contents

113. Find
Square of a Sum
Answer:
Simplify.
Example 8-1a

114. Check Check your work by using the FOIL method.
Example 8-1b

115. Find
Square of a Sum
Answer:
Simplify.
Example 8-1c

116. Find each product. a. b.
Answer:
Answer:
Example 8-1d

117. Find
Square of a Difference
Answer:
Simplify.
Example 8-2a

118. Find
Square of a Difference
Answer:
Simplify.
Example 8-2b

119. Find each product. a. b.
Answer:
Answer:
Example 8-2c

120. The formula for the area of a square is
Geometry Write an expression that represents the area of a square that has a side length of units.
The formula for the area of a square is
Area of a square
Simplify.
Answer: The area of the square is square units.
Example 8-3a

121. Geometry Write an expression that represents the area of a square that has a side length of units.
Answer:
Example 8-3b

122. Product of a Sum and a Difference
Find
Product of a Sum and a Difference
Answer:
Simplify.
Example 8-4a

123. Product of a Sum and a Difference
Find
Product of a Sum and a Difference
Answer:
Simplify.
Example 8-4b

124. Find each product. a. b.
Answer:
Answer:
Example 8-4c

125. End of Lesson 8

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