2. Welcome to Interactive Chalkboard Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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

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

6. Multiplying Monomials A monomial is a number, a variable, or a product of one or more variables. Monomials that are numbers without variables are called constants.

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

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

9. Rules for Multiplying Monomials Product of Powers: to multiply 2 powers that have the same base, add the exponents. Power of a Power: to find the power of a power, multiply the exponents. Power of a Product: To find the power of a product, find the power of each factor and then multiply.

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

11. Example 1-2b Simplify. Communicative and Associative Properties Product of Powers Simplify. Answer:

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

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

14. Example 1-3b Simplify Answer:

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

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

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

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

19. Example 1-5c Simplify Answer:

20. Monomial Review

21. End of Lesson 1

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

23. Simplifying monomials with divisionSimplifying monomials with division

24. Dividing Monomials Quotient of Powers: to divide two powers that have the same base, subtract the exponents. Power of a Quotient: to find the power of a quotient, find the power of the numerator and the power of the denominator. Zero Exponent: Any nonzero number raised to the zero power is equal to 1.

25. Negative Exponents:

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

27. Example 2-1b SimplifyAssume that a and b are not equal to zero. Answer:

28. Example 2-2a SimplifyAssume that e and f are not equal to zero. Power of a Quotient Power of a Product Power of a PowerAnswer:

29. Example 2-2b SimplifyAssume that p and q are not equal to zero. Answer:

30. Example 2-3a SimplifyAssume that m and n are not equal to zero. Answer: 1

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

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

33. Zero and Negative Exponents

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

35. Example 2-4b 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

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

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

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

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

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

41. Example 2-5d Answer: A Multiple-Choice Test Item Write the ratio of the circumference of the circle to the perimeter of the square in simplest form. AB CD

42. End of Lesson 2

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

44. Scientific Notation a X 10 n a must be between 1 and 10. n is an integer.

45. *Remember to count decimal carefully when converting numbers between scientific and standard notation.

46. Example 3-1a Express in standard notation. move decimal point 3 places to the left. Answer: 0.00748

47. Scientific Notation

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

49. Example 3-1c Express each number in standard notation. a. b. Answer: 0.0316 Answer: 7610

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

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

52. Express each number in scientific notation. a. 458,000,000 b. 0.0000452 Example 3-2c Answer:

53. Example 3-3a 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:

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

55. Example 3-3c The average circulation for all U.S. daily newspapers in 2000 was 111.5 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

56. Example 3-3d The average circulation for all U.S. daily newspapers in 2000 was 111.5 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: USA Today: The New York Times:

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

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

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

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

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

62. End of Lesson 3

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

64. Polynomials A polynomial is a monomial or a sum of monomials. A binomial is the sum or difference of two monomials. A trinomial is the sum or difference of three monomials.

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

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

67. Example 4-2a Write a polynomial to represent the area of the green shaded region. WordsThe area of the shaded region is the area of the rectangle minus the area of the triangle. Variablesarea of the shaded region height of rectangle area of rectangle triangle area

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

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

70. Degree The degree of a monomial is the sum of the exponents of all of its variables. The degree of a polynomial is the greatest degree of any term in the polynomial.

71. c. b. a. Degree of Polynomial Degree of Each Term TermsPolynomial Example 4-3a Find the degree of each polynomial. 88 22, 1, 0 30, 1, 2, 3

72. c. b. a. Degree of Polynomial Degree of Each Term TermsPolynomial Example 4-3b Find the degree of each polynomial. 77, 6 42, 4, 3 3 2, 1, 3, 0

73. Ascending order: arranging the terms so that the degree is in order from smallest to largest. The constant number will be first. Descending order: arranging the terms so that the degree is in order from largest to smallest. The constant number will be last.

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

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

76. Example 4-4c Arrange the terms of each polynomial so that the powers of x are in ascending order. a. b. Answer:

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

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

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

80. End of Lesson 4

81. Lesson 5 Contents Example 1Add Polynomials Example 2Subtract Polynomials Example 3Subtract Polynomials

82. Adding and subtracting Polynomials To add or subtract, add or subtract only like terms. Add the coefficients of each like terms.

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

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

85. Example 5-1c Find Answer:

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

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

88. Example 5-2c Find Answer:

89. Example 5-3a 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.

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

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

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

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

94. Polynomial Review

95. End of Lesson 5

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

97. Multiply a Polynomial by a Monomial Use the distributive property to multiply each term by the monomial. Use the rules for multiplying monomials.

98. Example 6-1a Find Method 1 Horizontal Distributive PropertyMultiply.

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

100. Example 6-1c Find Answer:

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

102. Example 6-2b Simplify Answer:

103. Example 6-3a 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. WordsThe total cost is the sum of the admission, super ride costs, and regular ride costs. VariablesIfthe number of super rides, then is the number of regular rides. Let M be the amount of money Sarita spent at the park.

104. Example 6-3b Equation Amount of moneyequalsadmissionplus super ridestimes $3 per rideplus regular ridestimes $2 per ride. M 10s3 2 Answer:An expression for the amount of money Sarita spent in the park is, where s is the number of super rides she rode. Distributive Property Simplify Simplify.

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

106. Example 6-3d 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: $21,200 Answer:

107. Solving Equations Use distributive property to eliminate ( ). Combine like terms. Solve the equation.

108. Example 6-4a Solve Subtractfrom each side. Original equation Distributive Property Combine like terms.

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

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

111. Example 6-4d Solve Answer:

112. End of Lesson 6

113. Lesson 7 Contents Example 1The Distributive Property Example 2FOIL Method Example 3FOIL Method Example 4The Distributive Property

114. Multiplying Polynomials Use the distributive property to multiply every term in both polynomials. FOIL Method: F: Multiply first terms O: Multiply outside terms I: Multiply inside terms L: Multiply last terms

115. Algebra Tiles Multiplying Binomials

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

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

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

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

120. Example 7-1e Find Answer:

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

122. Example 7-2b Find Multiply. Answer: Combine like terms. FIOL

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

124. Example 7-3a 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. Ahb

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

126. Example 7-3c Combine like terms. Distributive PropertyAnswer:The area of the triangle is square units.

127. Example 7-3d 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:

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

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

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

131. End of Lesson 7

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

133. Squaring a Binomial

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

135. Example 8-1b CheckCheck your work by using the FOIL method. F OIL

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

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

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

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

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

141. Example 8-3a 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.

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

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

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

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

146. End of Lesson 8

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