2. CHAPTER OUTLINE 10 Exponents and Polynomials Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10.1Addition and Subtraction of Polynomials 10.2Multiplication Properties of Exponents 10.3Multiplication of Polynomials 10.4Introduction to Factoring 10.5Negative Exponents and the Quotient Rule for Exponents 10.6Scientific Notation

3. Section Objectives 10.1 Addition and Subtraction of Polynomials Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Key Definitions 2.Addition of Polynomials 3.Subtraction of Polynomials 4.Evaluating Polynomials and Applications

4. Section 10.1 Addition and Subtraction of Polynomials 1.Key Definitions Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that a term is a number or a product or quotient of numbers and variables. A term in which the variables appear only in the numerator with whole number exponents is called a monomial. A polynomial is one or more monomials combined by addition or subtraction.

5. DEFINITION Categorizing Polynomials Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. If a polynomial has exactly one term, then it is called a monomial. Example: 3xy 4 (1 term) If a polynomial has exactly two terms, then it is called a binomial. Example: 5ab + 6 (2 terms) If a polynomial has exactly three terms, then it is called a trinomial. Example: 6x 4 – 7x 2 – 5x (3 terms)

6. Section 10.1 Addition and Subtraction of Polynomials 1.Key Definitions Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. This is written in descending order. The degree is 6.

7. Example 1Identifying the Characteristics of a Polynomial Slide 7 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write each polynomial in descending order. Determine the degree of the polynomial, and categorize the polynomial as a monomial, binomial, or trinomial.

8. Example Solution: 1Identifying the Characteristics of a Polynomial Slide 8 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

9. Section 10.1 Addition and Subtraction of Polynomials 1.Key Definitions Slide 9 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The coefficient of a term is the numerical factor of the term.

10. Section 10.1 Addition and Subtraction of Polynomials 2.Addition of Polynomials Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Like terms have the same variables, raised to the same powers.

11. Example 2Combining Like Terms Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12. Example Solution: 2Combining Like Terms Slide 12 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13. Section 10.1 Addition and Subtraction of Polynomials 2.Addition of Polynomials Slide 13 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To add two polynomials, first use the associative and commutative properties of addition to group like terms. Then combine like terms.

14. Example 3Adding Polynomials Slide 14 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

15. Example Solution: 3Adding Polynomials Slide 15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16. TIP: Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Polynomials can also be added vertically. Begin by lining up like terms in the same column.

17. Section 10.1 Addition and Subtraction of Polynomials 3.Subtraction of Polynomials Slide 17 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that subtraction of real numbers is defined as a – b = a + (–b). That is, we add the opposite of the second number to the first number. We will use the same strategy to subtract polynomials. To find the opposite of a polynomial, take the opposite of each term.

18. Example 5Finding the Opposite of a Polynomial Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the opposite of the polynomial.

19. Example Solution: 5Finding the Opposite of a Polynomial Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The opposite of a real number a is written as (–a). We can apply the same notation to find the opposite of a polynomial. The opposite of 2x 2 – 7x + 5 is –(2x 2 – 7x + 5). = –2x 2 + 7x – 5Apply the distributive property.

20. Example 6Subtracting Polynomials Slide 20 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21. Example Solution: 6Subtracting Polynomials Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the opposite of the second polynomial by applying the distributive property. Regroup and collect like terms. Combine like terms.

22. TIP: Slide 22 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtraction of polynomials can also be performed vertically. To do so, add the opposite of the second polynomial to the first polynomial.

23. Example 7Subtracting Polynomials Slide 23 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

24. Example Solution: 7Subtracting Polynomials Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25. Section 10.1 Addition and Subtraction of Polynomials 4.Evaluating Polynomials and Applications Slide 25 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A polynomial is an algebraic expression. Evaluating a polynomial for a value of the variable is the same as evaluating an expression.

26. Example 8Evaluating a Polynomial Slide 26 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Evaluate the polynomial for the given value of the variable:

27. Example Solution: 8Evaluating a Polynomial Slide 27 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

28. Example 9Evaluating a Polynomial in an Application Slide 28 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The cost (in dollars) to rent storage space for x months is given by the polynomial. 49.99x + 129 a. Evaluate the polynomial for x = 3 and interpret the result in the context of the problem. b. Evaluate the polynomial for x = 12 and interpret the result in the context of the problem.

29. Example Solution: 9Evaluating a Polynomial in an Application Slide 29 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

30. Section Objectives 10.2 Multiplication Properties of Exponents Slide 30 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Multiplication of Like Bases: a m a n = a mn 2.Multiplying Monomials 3.Power Rule of Exponents: (a m ) n = a m n 4.The Power of a Product and the Power of a Quotient

31. PROPERTYMultiplication of Factors with Like Bases Slide 31 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

32. Example 1Multiplying Factors with the Same Base Slide 32 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

33. Example Solution: 1Multiplying Factors with the Same Base Slide 33 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The base remains unchanged. Add the exponents.

34. Section 10.2 Multiplication Properties of Exponents 2.Multiplying Monomials Slide 34 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We use the commutative and associative properties of multiplication to regroup factors and multiply like bases.

35. Example 2Multiplying Monomials Slide 35 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

36. Example Solution: 2Multiplying Monomials Slide 36 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

37. Example 3Multiplying Monomials Slide 37 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

38. Example Solution: 3Multiplying Monomials Slide 38 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

39. PROPERTYPower Rule for Exponents Slide 39 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

40. Example 4Applying the Power Rule for Exponents Slide 40 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Simplify the expressions.

41. Example Solution: 4Applying the Power Rule for Exponents Slide 41 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

42. Avoiding Mistakes Slide 42 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

43. PROPERTYPower of a Product and the Power of a Quotient Slide 43 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

44. Example 5Simplifying a Power of a Product or Quotient Slide 44 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Simplify.

45. Example Solution: 5Simplifying a Power of a Product or Quotient Slide 45 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

46. Example 6Simplifying Expressions Involving Exponents Slide 46 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

47. Example Solution: 6Simplifying Expressions Involving Exponents Slide 47 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Apply the power rule to each factor in parentheses. Operations with exponents are performed before multiplication.

48. Section Objectives 10.3 Multiplication of Polynomials Slide 48 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Multiplying a Monomial by a Polynomial 2.Multiplying a Polynomial by a Polynomial

49. Section 10.3 Multiplication of Polynomials 1.Multiplying a Monomial by a Polynomial Slide 49 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To multiply a monomial by a polynomial with more than one term, use the distributive property.

50. Example 1Multiplying a Monomial by a Polynomial Slide 50 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

51. Example Solution: 1Multiplying a Monomial by a Polynomial Slide 51 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

52. Example 2Multiplying a Monomial by a Polynomial Slide 52 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

53. Example Solution: 2Multiplying a Monomial by a Polynomial Slide 53 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

54. PROCEDUREMultiplying Polynomials Slide 54 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 To multiply two polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. Step 2 Combine like terms if possible.

55. Example 4Multiplying a Binomial by a Binomial Slide 55 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

56. Example Solution: 4Multiplying a Binomial by a Binomial Slide 56 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multiply each term in the first binomial by each term in the second binomial.

57. TIP: Slide 57 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notice that the product of two binomials equals the sum of the products of the First, Outer, Inner, and Last terms. The word “FOIL” can be used as a memory device to multiply two binomials. Note that FOIL only works when multiplying binomials.

58. Example 5Multiplying a Binomial by a Binomial Slide 58 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

59. Example Solution: 5Multiplying a Binomial by a Binomial Slide 59 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multiply each term in the first binomial by each term in the second binomial. Simplify each term. Notice that there are no like terms to combine.

60. Example 7Squaring a Binomial Slide 60 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

61. Example Solution: 7Squaring a Binomial Slide 61 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To square a quantity, multiply the quantity times itself.

62. Example 8Multiplying a Polynomial by a Polynomial Slide 62 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

63. Example Solution: 8Multiplying a Polynomial by a Polynomial Slide 63 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Apply the distributive property. Multiply each term in the first polynomial by each term in the second.

64. Section Objectives 10.4 Introduction to Factoring Slide 64 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Greatest Common Factor 2.Factoring Out the Greatest Common Factor

65. Section 10.4 Introduction to Factoring 1.Greatest Common Factor Slide 65 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The greatest common factor (denoted GCF) of two or more integers is the greatest factor that divides evenly into each integer.

66. PROCEDUREFinding the GCF of Two or More Integers Slide 66 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Factor each integer into prime factors. Step 2 Determine the prime factors common to each integer, including repeated factors. Step 3 The product of the factors from step 2 is the GCF

67. Example 1Determining the Greatest Common Factor Slide 67 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine the GCF.

68. Example Solution: 1Determining the Greatest Common Factor (continued) Slide 68 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

69. Example Solution: 1Determining the Greatest Common Factor Slide 69 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

70. Example 2Determining the Greatest Common Factor Slide 70 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine the GCF.

71. Example Solution: 2Determining the Greatest Common Factor (continued) Slide 71 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

72. Example Solution: 2Determining the Greatest Common Factor Slide 72 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

73. Example 3Factoring Out the GCF Slide 73 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

74. Example Solution: 3Factoring Out the GCF Slide 74 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

75. Example 5Factoring Out the GCF Slide 75 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

76. Example Solution: 5Factoring Out the GCF Slide 76 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

77. Section Objectives 10.5 Negative Exponents and the Quotient Rule for Exponents Slide 77 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Division of Like Bases 2.Definition of b 0 3.Definition of b –n 4.Properties of Exponents: A Summary

78. PROPERTYDivision of Like Bases Slide 78 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

79. Example 1Dividing Like Bases Slide 79 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

80. Example Solution: 1Dividing Like Bases Slide 80 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The base is unchanged. Subtract the exponents.

81. DEFINITIONb0b0 Slide 81 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let b be a nonzero number. Then b 0 = 1.

82. Example 2Simplifying Expressions with a Zero Exponent Slide 82 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

83. Example Solution: 2Simplifying Expressions with a Zero Exponent Slide 83 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

84. DEFINITIONb –n Slide 84 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

85. Section 10.5 Negative Exponents and the Quotient Rule for Exponents 3.Definition of b –n Slide 85 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To evaluate b –n, take the reciprocal of the base and change the sign of the exponent.

86. Example 3Simplifying Expressions Containing Negative Exponents Slide 86 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

87. Example Solution: 3Simplifying Expressions Containing Negative Exponents (continued) Slide 87 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Take the reciprocal of the base. Change the sign of the exponent.

88. Example Solution: 3Simplifying Expressions Containing Negative Exponents Slide 88 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

89. Example 4Simplifying Expressions Containing Negative Exponents Slide 89 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

90. Example Solution: 4Simplifying Expressions Containing Negative Exponents (continued) Slide 90 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Take the reciprocal of the base. Change the sign of the exponent. Square the numerator and square the denominator.

91. Example Solution: 4Simplifying Expressions Containing Negative Exponents Slide 91 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. There are no parentheses to group the 5 and x as a single base. Therefore, the exponent of –2 applies only to x. The factor of 5 has an implied exponent of 1.

92. Example 5Simplifying Expressions Containing Negative Exponents Slide 92 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

93. Example Solution: 5Simplifying Expressions Containing Negative Exponents Slide 93 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The negative exponent on y changes the position of y within the fraction. Notice that x does not change position because its exponent is positive. The factor of a and the factor of c have negative exponents. Change their positions within the fraction. Notice that b does not change position because its exponent is positive.

94. Section 10.5 Negative Exponents and the Quotient Rule for Exponents 4.Properties of Exponents: A Summary Slide 94 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

95. Example 6Simplifying Expressions Containing Exponents Slide 95 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

96. Example Solution: 6Simplifying Expressions Containing Exponents (continued) Slide 96 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The base is unchanged. Add the exponents (property 1). The base is unchanged. Subtract the exponents (property 2).

97. Example Solution: 6Simplifying Expressions Containing Exponents Slide 97 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

98. Section Objectives 10.6 Scientific Notation Slide 98 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Scientific Notation 2.Converting to Scientific Notation 3.Converting Scientific Notation to Standard Form

99. Section 10.6 Scientific Notation 1.Scientific Notation Slide 99 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Scientific notation is a means by which we can write very large numbers and very small numbers without having to write numerous zeros in the number.

100. Section 10.6 Scientific Notation 1.Scientific Notation Slide 100 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Using scientific notation we express a number as the product of two factors. One factor is a number greater than or equal to 1, but less than 10. The other factor is a power of 10.

101. DEFINITIONScientific Notation Slide 101 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A positive number written in scientific notation is written as a 10 n, where a is a number greater than or equal to 1, but less than 10, and n is an integer.

102. Section 10.6 Scientific Notation 2.Converting to Scientific Notation (continued) Slide 102 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To write a number in scientific notation, follow these guidelines. Move the decimal point so that its new location is to the right of the first nonzero digit. Count the number of places that the decimal point is moved. Then 1. If the original number is greater than or equal to 10:

103. Section 10.6 Scientific Notation 2.Converting to Scientific Notation (continued) Slide 103 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The exponent for the power of 10 is positive and is equal to the number of places that the decimal point was moved.

104. Section 10.6 Scientific Notation 2.Converting to Scientific Notation (continued) Slide 104 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2. If the original number is between 0 and 1: The exponent for the power of 10 is negative. Its absolute value is equal to the number of places the decimal point was moved.

105. Section 10.6 Scientific Notation 2.Converting to Scientific Notation Slide 105 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For this case, scientific notation is not needed. 3. If the original number is between 1 and 10: The exponent on 10 is 0.

106. Example 1Writing Numbers in Scientific Notation Slide 106 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the number in scientific notation. a. 93,000,000 mi (the distance between Earth and the Sun) b. 0.000 000 000 753 kg (the mass of a dust particle) c. 300,000,000 m/sec (the speed of light) d. 0.00017 m (length of the smallest insect in the world)

107. Example Solution: 1Writing Numbers in Scientific Notation (continued) Slide 107 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The number is greater than 10. Move the decimal point left 7 places. For a number greater than 10, the exponent is positive. The number is between 0 and 1. Move the decimal point to the right 10 places. For a number between 0 and 1, the exponent is negative.

108. Example Solution: 1Writing Numbers in Scientific Notation Slide 108 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

109. Section 10.6 Scientific Notation 3.Converting Scientific Notation to Standard Form Slide 109 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To convert from scientific notation to standard form, follow these guidelines. 1. If the exponent on 10 is positive, move the decimal point to the right the same number of places as the exponent. Add zeros as necessary. 2. If the exponent on 10 is negative, move the decimal point to the left the same number of places as the exponent. Add zeros as necessary.

110. Example 2Converting Scientific Notation to Standard Form Slide 110 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convert to decimal notation.

111. Example Solution: 2Converting Scientific Notation to Standard Form Slide 111 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.