1. Copyright © 2011 Pearson Education, Inc. Polynomials CHAPTER 5.1Exponents and Scientific Notation 5.2Introduction to Polynomials 5.3Adding and Subtracting Polynomials 5.4Exponent Rules and Multiplying Monomials 5.5Multiplying Polynomials; Special Products 5.6Exponent Rules and Dividing Polynomials 5

2. Copyright © 2011 Pearson Education, Inc. Exponents and Scientific Notation 5.1 1.Evaluate exponential forms with integer exponents. 2.Write scientific notation in standard form. 3.Write standard form numbers in scientific notation.

3. Slide 5- 3 Copyright © 2011 Pearson Education, Inc. Objective 1 Evaluate exponential forms with integer exponents.

4. Slide 5- 4 Copyright © 2011 Pearson Education, Inc. Evaluating Exponential Forms with Negative Bases If the base of an exponential form is a negative number and the exponent is even, then the product is positive. If the base is a negative number and the exponent is odd, then the product is negative. Raising a Quotient to a Power If a and b are real numbers, where b 0 and n is a natural number, then

5. Slide 5- 5 Copyright © 2011 Pearson Education, Inc. Example 1 Evaluate each exponential form. a. b.c. Solution a. b. c.

6. Slide 5- 6 Copyright © 2011 Pearson Education, Inc. Zero as an Exponent If a is a real number and then If a is a real number, where a 0 and n is a natural number, then

7. Slide 5- 7 Copyright © 2011 Pearson Education, Inc. Example 2 Rewrite with a positive exponent; then if the expression is numeric, evaluate it. a. b.c. Solution a. b. c. Rewrite using then simplify.

8. Slide 5- 8 Copyright © 2011 Pearson Education, Inc. continued d. e. f. Solution d. e. f.

9. Slide 5- 9 Copyright © 2011 Pearson Education, Inc. If a is a real number, where a 0 and n is a natural number, then If a and b are real numbers, where a 0 and b 0 and n is a natural number, then

10. Slide 5- 10 Copyright © 2011 Pearson Education, Inc. Example 3 Rewrite with a positive exponent; then if the expression is numeric, evaluate it. a.b. Solution a. b.

11. Slide 5- 11 Copyright © 2011 Pearson Education, Inc. Example 4 Rewrite with a positive exponent, then if the expression is numeric, evaluate it. a.b. Solution a. b.

12. Slide 5- 12 Copyright © 2011 Pearson Education, Inc. Objective 2 Write scientific notation in standard form.

13. Slide 5- 13 Copyright © 2011 Pearson Education, Inc. Scientific notation: A number expressed in the form where a is a decimal number with and n is an integer. Scientific notation gives us a shorthand way to write very large or very small numbers. Changing Scientific Notation (Positive Exponent) to Standard Form To change from scientific notation with a positive integer exponent to standard form, move the decimal point to the right the number of places indicated by the exponent.

14. Slide 5- 14 Copyright © 2011 Pearson Education, Inc. Example 5 Write in standard form. Solution Multiplying 4.26 by 10 5 means that the decimal point will move five places to the right.

15. Slide 5- 15 Copyright © 2011 Pearson Education, Inc. Changing Scientific Notation (Negative Exponent) to Standard Form To change from scientific notation with a negative exponent in standard form, move the decimal point to the left the same number of places as the absolute value of the exponent.

16. Slide 5- 16 Copyright © 2011 Pearson Education, Inc. Example 6 Write in standard form. Solution Multiplying 3.87 by 10 –4 is equivalent to dividing by 10 4, which causes the decimal point to move four places to the left.

17. Slide 5- 17 Copyright © 2011 Pearson Education, Inc. Objective 3 Write standard form numbers in scientific notation.

18. Slide 5- 18 Copyright © 2011 Pearson Education, Inc. Changing Standard Form to Scientific Notation To write a number greater than 1 in scientific notation: 1. Move the decimal point so that the number is greater than or equal to 1 but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.) 2. Write the decimal number multiplied by 10 n, where n is the number of places between the new decimal position and the original decimal position. 3. Delete zeros to the right of the last nonzero digit.

19. Slide 5- 19 Copyright © 2011 Pearson Education, Inc. Example 7 Write 875,000 in scientific notation. Solution Place the decimal to the right of the first nonzero digit, 8. Next, we count the places to the right of this decimal position, which is five places. We write the 5 as the exponent of 10. Finally, we delete all the 0s to the right of the last nonzero digit, which is 5 in this case. Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10. There are five places to the right of the new decimal point.

20. Slide 5- 20 Copyright © 2011 Pearson Education, Inc. Changing Standard Form to Scientific Notation To write a positive decimal number that is less than 1 in scientific notation: 1. Move the decimal point so that the number is greater than or equal to 1 but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.) 2. Write the decimal number multiplied by 10 n, where n is a negative integer whose absolute value is the number of places between the new decimal position and the original decimal position. 3.Delete zeros to the left of the first nonzero digit.

21. Slide 5- 21 Copyright © 2011 Pearson Education, Inc. Example 8 Write 0.0000000472 in scientific notation. Solution Place the decimal point between the 4 and 7 digits, so that we have 4.72, which is a decimal number greater than 1 but less than 10. Since there are eight decimal places between the original decimal position and the new position, the exponent is –8. Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10. There are eight decimal places in between the original position and the new position, so the exponent is –8.

22. Slide 5- 22 Copyright © 2011 Pearson Education, Inc. Evaluate –2 –6. a) b) c) d) 64 5.1

23. Slide 5- 23 Copyright © 2011 Pearson Education, Inc. Evaluate –2 –6. a) b) c) d) 64 5.1

24. Slide 5- 24 Copyright © 2011 Pearson Education, Inc. Write 13,030,000 in scientific notation. a) b) c) d) 5.1

25. Slide 5- 25 Copyright © 2011 Pearson Education, Inc. Write 13,030,000 in scientific notation. a) b) c) d) 5.1

26. Slide 5- 26 Copyright © 2011 Pearson Education, Inc. Write in standard notation. a) 4,690,000 b) 4,690,000,000 c) 0.000000469 d) 0.00000469 5.1

27. Slide 5- 27 Copyright © 2011 Pearson Education, Inc. Write in standard notation. a) 4,690,000 b) 4,690,000,000 c) 0.000000469 d) 0.00000469 5.1

28. Copyright © 2011 Pearson Education, Inc. Introduction to Polynomials 5.2 1.Identify monomials. 2.Identify the coefficient and degree of a monomial. 3.Classify polynomials. 4.Identify the degree of a polynomial. 5.Evaluate polynomials. 6.Write polynomials in descending order of degree. 7.Combine like terms.

29. Slide 5- 29 Copyright © 2011 Pearson Education, Inc. Objective 1 Identify monomials.

30. Slide 5- 30 Copyright © 2011 Pearson Education, Inc. Monomial: An expression that is a constant, a variable, or a product of a constant and variable(s) that are raised to whole number powers.

31. Slide 5- 31 Copyright © 2011 Pearson Education, Inc. Example 1 Is the given expression a monomial? Explain. a. 18b. –0.4a 2 bc. 5a 2 + 4b – 1 Answer a. 18 is a monomial because it is a constant. b. –0.4a 2 b is a monomial because it is a product of a constant, –0.4, and variables, a 2 and b, which have whole-number exponents. c. 5a 2 + 4b – 1 is not a monomial because it is not a product of a constant and variables. Instead, addition and subtraction are involved.

32. Slide 5- 32 Copyright © 2011 Pearson Education, Inc. Objective 2 Identify the coefficient and degree of a monomial.

33. Slide 5- 33 Copyright © 2011 Pearson Education, Inc. Coefficient of a monomial: The numerical factor in a monomial. Degree of a monomial: The sum of the exponents of all variables in a monomial.

34. Slide 5- 34 Copyright © 2011 Pearson Education, Inc. Example 2 Identify the coefficient and degree of each monomial. a. b. 9 Answer a. We can express as. In this form, we can see that our coefficient is –2 and the exponents for the variables are 1 and 3. Since the degree is the sum of the variables’ exponents, the degree is 4. b. Since 9 = 9x 0, where x is any real number except 0, we can see that 9 is the coefficient and 0 is the degree.

35. Slide 5- 35 Copyright © 2011 Pearson Education, Inc. Objective 3 Classify polynomials.

36. Slide 5- 36 Copyright © 2011 Pearson Education, Inc. Polynomial: A monomial or an expression that can be written as a sum of monomials. Examples: 4x, 4x + 8, 2x 2 + 5xy + 8y Polynomial in one variable: A polynomial in which every variable term has the same variable. Example: x 2 – 5x + 2 is a polynomial in one variable Binomial: A polynomial containing two terms. Trinomial: A polynomial containing three terms. Degree of a polynomial: The greatest degree of any of the terms in the polynomial.

37. Slide 5- 37 Copyright © 2011 Pearson Education, Inc. Example 3 Determine whether the expression is a monomial, binomial, trinomial, or none of these. a. 4ab 2 b. –9x 2 + zc. 4n 3 + 2n – 1 Answer a. 4ab 2 is a monomial because it has a single term. b. –9x 2 + z is a binomial because it contains two terms. c. 4n 3 + 2n – 1 is a trinomial because it contains three terms.

38. Slide 5- 38 Copyright © 2011 Pearson Education, Inc. continued Determine whether the expression is a monomial, binomial, trinomial, or none of these. d. x 3 + 9x 2 – x + 4 e. Answer d. Although x 3 + 9x 2 – x + 4 is a polynomial, it has no special name because it has more than three terms. e. is not a polynomial because is not a monomial.

39. Slide 5- 39 Copyright © 2011 Pearson Education, Inc. Objective 4 Identify the degree of a polynomial.

40. Slide 5- 40 Copyright © 2011 Pearson Education, Inc. Example 4a Identify the degree of the polynomial. Answer The degree is 6 because it is the greatest degree of all the terms.

41. Slide 5- 41 Copyright © 2011 Pearson Education, Inc. Example 4b Identify the degree of the polynomial. Answer To determine the degree of the monomial –2a 5 b 2, add the exponents of its variables: 5 + 2 = 7 Add the exponents of the variables in ab 2 : 1 + 2 = 3 Compare the degrees of all the terms: 4, 7, 3, 1 7 is the greatest degree. Therefore, 7 is the degree of the polynomial.

42. Slide 5- 42 Copyright © 2011 Pearson Education, Inc. Objective 5 Evaluate polynomials.

43. Slide 5- 43 Copyright © 2011 Pearson Education, Inc. Example 5 Evaluate when c = –6. Solution Replace c with –6. Simplify.

44. Slide 5- 44 Copyright © 2011 Pearson Education, Inc. Example 6 If we neglect air resistance, the polynomial −16t 2 + h 0 describes the height of a falling object from an initial height h 0 for t seconds. If a rock is dropped from a building that is 120 feet tall, what is the height of the rock after it falls for 2 seconds? Answer Evaluate −16t 2 + h 0 when t = 2 and h 0 = 120. −16(2) 2 + 120 = 56 = −64 + 120 After 2 seconds, the height of the rock is 56 feet.

45. Slide 5- 45 Copyright © 2011 Pearson Education, Inc. Objective 6 Write polynomials in descending order of degree.

46. Slide 5- 46 Copyright © 2011 Pearson Education, Inc. Writing a Polynomial in Descending Order of Degree To write a polynomial in descending order of degree, place the highest degree term first, then the next highest degree, and so on.

47. Slide 5- 47 Copyright © 2011 Pearson Education, Inc. Example 7 Write the polynomial in descending order. Solution Rearrange the terms so that the highest degree term is first, then the next highest degree, and so on. Degree 4Degree 3Degree 2Degree 1Degree 0 Answer

48. Slide 5- 48 Copyright © 2011 Pearson Education, Inc. Objective 7 Combine like terms.

49. Slide 5- 49 Copyright © 2011 Pearson Education, Inc. Example 8 Combine like terms and write the resulting polynomial in descending order of degree. Solution

50. Slide 5- 50 Copyright © 2011 Pearson Education, Inc. Example 9 Combine like terms. Solution

51. Slide 5- 51 Copyright © 2011 Pearson Education, Inc. continued Alternative Solution Instead of first collecting like terms, we strike through like terms in the given polynomial as they are combined.

52. Slide 5- 52 Copyright © 2011 Pearson Education, Inc. Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these 5.2

53. Slide 5- 53 Copyright © 2011 Pearson Education, Inc. Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these 5.2

54. Slide 5- 54 Copyright © 2011 Pearson Education, Inc. Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2

55. Slide 5- 55 Copyright © 2011 Pearson Education, Inc. Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2

56. Slide 5- 56 Copyright © 2011 Pearson Education, Inc. Identify the degree of the polynomial. a) 3 b) 5 c) 6 d) 7 5.2

57. Slide 5- 57 Copyright © 2011 Pearson Education, Inc. Identify the degree of the polynomial. a) 3 b) 5 c) 6 d) 7 5.2

58. Copyright © 2011 Pearson Education, Inc. Adding and Subtracting Polynomials 5.3 1.Add polynomials. 2.Subtract polynomials.

59. Slide 5- 59 Copyright © 2011 Pearson Education, Inc. Objective 1 Add polynomials.

60. Slide 5- 60 Copyright © 2011 Pearson Education, Inc. We can add and subtract polynomials in the same way that we add and subtract numbers. In fact, polynomials are like whole numbers that are in an expanded form. In our base-ten number system, each place value is a power of 10. We can think of polynomials as a variable-base number system, where x 2 is like the hundreds place (10 2 ) and x is like the tens place (10 1 ). To add whole numbers, we add the digits in like place values; in polynomials, we add like terms. Adding Polynomials To add polynomials, combine like terms.

61. Slide 5- 61 Copyright © 2011 Pearson Education, Inc. Example 1 Add and write the resulting polynomial in descending order of degree. Solution

62. Slide 5- 62 Copyright © 2011 Pearson Education, Inc. Example 2 Add and write the resulting polynomial in descending order of degree. Solution

63. Slide 5- 63 Copyright © 2011 Pearson Education, Inc. Example 3 Write an expression in simplest form for the perimeter of the rectangle shown. 5b + 2 9b – 10 Understand Perimeter means the total distance around the shape. Therefore, we need to add the lengths of all the sides. Plan The lengths of the sides are represented by polynomials. Therefore, we add the polynomials to represent the perimeter.

64. Slide 5- 64 Copyright © 2011 Pearson Education, Inc. continued Execute Perimeter = Length + Width + Length + Width Answer The expression for the perimeter is.

65. Slide 5- 65 Copyright © 2011 Pearson Education, Inc. continued Check To check: 1. Choose a value for b and evaluate the original expressions for length and width. 2. Determine the corresponding numeric perimeter. 3. Evaluate the perimeter expression using the same value for b and verify that we get the same numeric perimeter. Let’s choose b = 2. Length:Width: Perimeter 408 8 12 + + + =

66. Slide 5- 66 Copyright © 2011 Pearson Education, Inc. continued Now evaluate the perimeter expression where b = 2 and we should find that the result is 40. Perimeter expression: This agrees with our calculation above.

67. Slide 5- 67 Copyright © 2011 Pearson Education, Inc. Objective 2 Subtract polynomials.

68. Slide 5- 68 Copyright © 2011 Pearson Education, Inc. Subtracting Polynomials To subtract polynomials, 1. Write the subtraction statement as an equivalent addition statement. a. Change the operation symbol from a minus sign to a plus sign. b. Change the subtrahend (second polynomial) to its additive inverse. To get the additive inverse, we change the sign of each term in the polynomial. 2. Combine like terms.

69. Slide 5- 69 Copyright © 2011 Pearson Education, Inc. Example 4 Subtract. Solution Change the minus sign to a plus sign. Change all signs in the subtrahend.

70. Slide 5- 70 Copyright © 2011 Pearson Education, Inc. Example 5 Subtract. Solution Change the minus sign to a plus sign. Change all signs in the subtrahend. Subtract.

71. Slide 5- 71 Copyright © 2011 Pearson Education, Inc. Add and write the polynomial in descending order. a) b) c) d) 5.3

72. Slide 5- 72 Copyright © 2011 Pearson Education, Inc. Add and write the polynomial in descending order. a) b) c) d) 5.3

73. Slide 5- 73 Copyright © 2011 Pearson Education, Inc. Subtract and write the polynomial in descending order. a) b) c) d) 5.3

74. Slide 5- 74 Copyright © 2011 Pearson Education, Inc. Subtract and write the polynomial in descending order. a) b) c) d) 5.3

75. Copyright © 2011 Pearson Education, Inc. Exponent Rules and Multiplying Monomials 5.4 1.Multiply monomials. 2.Multiply numbers in scientific notation. 3.Simplify a monomial raised to a power.

76. Slide 5- 76 Copyright © 2011 Pearson Education, Inc. Objective 1 Multiply monomials.

77. Slide 5- 77 Copyright © 2011 Pearson Education, Inc. Consider 2 3 2 4, which is a product of exponential forms. To simplify 2 3 2 4, we could follow the order of operations and evaluate the exponential forms first, then multiply. 2 3 2 4 = 8 16 = 128 However, there is an alternative. We can write the result in exponential form by first writing 2 3 and 2 4 in their factored forms. 2 3 means three 2s. 2 4 means four 2s. Since there are a total of seven 2s multiplied, we can express the product as 2 7. Notice that 2 7 = 128.

78. Slide 5- 78 Copyright © 2011 Pearson Education, Inc. Product Rule for Exponents If a is a real number and m and n are integers, then a m a n = a m+n. Multiplying Monomials To multiply monomials: 1. Multiply coefficients. 2. Add the exponents of the like bases. 3.Write any unlike variable bases unchanged in the product.

79. Slide 5- 79 Copyright © 2011 Pearson Education, Inc. Example 1 Multiply Solution Because the bases are the same, we can add the exponents and keep the same base.

80. Slide 5- 80 Copyright © 2011 Pearson Education, Inc. Example 2 Multiply. a. b. Solution a. b. 1 3 Multiply the coefficients and add the exponents of the like bases.

81. Slide 5- 81 Copyright © 2011 Pearson Education, Inc. Example 3 Write an expression in simplest form for the volume of the box shown. Understand We are given a box with side lengths that are monomial expressions. 3b3b 5b5b b Plan The volume of a box is found by multiplying the length, width, and height.

82. Slide 5- 82 Copyright © 2011 Pearson Education, Inc. continued Execute Answer The expression for volume is 15b 3. Check Since is an identity, substitute 2 for b and solve.

83. Slide 5- 83 Copyright © 2011 Pearson Education, Inc. Objective 2 Multiply numbers in scientific notation.

84. Slide 5- 84 Copyright © 2011 Pearson Education, Inc. We multiply numbers in scientific notation using the same procedure we used to multiply monomials. Monomials:Scientific notation:

85. Slide 5- 85 Copyright © 2011 Pearson Education, Inc. Example 4a Multiply. Write the answer in scientific notation. Solution Multiply 4.5 and 5.7, then add the exponents for base 10s. Note: The product is not in scientific notation. We must move the decimal point one place to the left and account for this by increasing the exponent by one.

86. Slide 5- 86 Copyright © 2011 Pearson Education, Inc. Example 4b Multiply. Write the answer in scientific notation. Solution Multiply 6.2 and 3.1, then add the exponents for base 10s. Note: The product is not in scientific notation. We must move the decimal point one place to the left and account for this by increasing the exponent by one.

87. Slide 5- 87 Copyright © 2011 Pearson Education, Inc. Objective 3 Simplify a monomial raised to a power.

88. Slide 5- 88 Copyright © 2011 Pearson Education, Inc. A Power Raised to a Power If a is a real number and m and n are integers, then (a m ) n = a mn. Raising a Product to a Power If a and b are real numbers and n is an integer, then (ab) n = a n b n. Simplifying a Monomial Raised to a Power To simplify a monomial raised to a power, 1. Evaluate the coefficient raised to that power. 2. Multiply each variable’s exponent by the power.

89. Slide 5- 89 Copyright © 2011 Pearson Education, Inc. Example 5 Simplify. a.b.c. Solution a.

90. Slide 5- 90 Copyright © 2011 Pearson Education, Inc. Example 6a Simplify. Solution Since the order of operations is to simplify exponents before multiplying, we will simplify (8y 7 ) 3 first, then multiply the result by 6y 2. Multiply coefficients and add exponents of like variables. Simplify.

91. Slide 5- 91 Copyright © 2011 Pearson Education, Inc. Example 6b Simplify. Solution We follow the order of operations and simplify the monomials raised to a power first, then multiply the monomials. Multiply coefficients and add exponents of like variables. Simplify.

92. Slide 5- 92 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.4

93. Slide 5- 93 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.4

94. Slide 5- 94 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.4

95. Slide 5- 95 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.4

96. Copyright © 2011 Pearson Education, Inc. Multiplying Polynomials; Special Products 5.5 1.Multiply a polynomial by a monomial. 2.Multiply binomials. 3. Multiply polynomials. 4.Determine the product when given special polynomial factors.

97. Slide 5- 97 Copyright © 2011 Pearson Education, Inc. Objective 1 Multiply a polynomial by a monomial.

98. Slide 5- 98 Copyright © 2011 Pearson Education, Inc. Multiplying a Polynomial by a Monomial To multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial.

99. Slide 5- 99 Copyright © 2011 Pearson Education, Inc. Example 1 Multiply. a. Solution a. 2p 6p 2 2p 2p –1

100. Slide 5- 100 Copyright © 2011 Pearson Education, Inc. continued b. Solution b. Note: When multiplying multivariable terms, it is helpful to multiply the coefficients first, then the variables in alphabetical order. –2a 2 b 3a 3 b –2a 2 b ab 3 –2a 2 b – 5a 4 –2a 2 b 4bc

101. Slide 5- 101 Copyright © 2011 Pearson Education, Inc. Objective 2 Multiply binomials.

102. Slide 5- 102 Copyright © 2011 Pearson Education, Inc. Multiplying Polynomials To multiply two polynomials, 1. Multiply every term in the second polynomial by every term in the first polynomial. 2. Combine like terms.

103. Slide 5- 103 Copyright © 2011 Pearson Education, Inc. Example 2 Multiply. Solution Multiply each term in x + 7 by each term in x + 3. 7 x 7 3 x 3 x

104. Slide 5- 104 Copyright © 2011 Pearson Education, Inc. Example 3 Multiply. Solution Multiply each term in x – 5 by each term in 2x + 1 (think FOIL). FirstOuterInner Last 1 x Inner 1 (–5) Last 2x (–5) Outer 2x x First

105. Slide 5- 105 Copyright © 2011 Pearson Education, Inc. continued

106. Slide 5- 106 Copyright © 2011 Pearson Education, Inc. Example 4 Multiply. Solution Multiply each term in 2x + 7 by each term in 3x – 5 (think FOIL). 3x 2x 3x 7 (–5) 2x (–5) 7 FirstOuterInnerLast

107. Slide 5- 107 Copyright © 2011 Pearson Education, Inc. continued

108. Slide 5- 108 Copyright © 2011 Pearson Education, Inc. Example 5 Multiply. Solution Multiply each term in 4x − 2 by each term in 2x – 3 (think FOIL). 4x 2x 4x (–3) (–2) 2x (–2) (–3) FirstOuterInner Last

109. Slide 5- 109 Copyright © 2011 Pearson Education, Inc. continued

110. Slide 5- 110 Copyright © 2011 Pearson Education, Inc. The product of two binomials can be shown in terms of geometry. 35 5x5x 7x7xx2x2 Length width = Sum of the areas of the four internal rectangles Combine like terms.

111. Slide 5- 111 Copyright © 2011 Pearson Education, Inc. Objective 3 Multiply polynomials.

112. Slide 5- 112 Copyright © 2011 Pearson Education, Inc. Example 6 Multiply. Solution Multiply each term in 2x 2 + 3x +3 by each term in x – 3. (–3) 2x 2 (–3) 3x (–3) 3 x 2x 2 x 3x x 3

113. Slide 5- 113 Copyright © 2011 Pearson Education, Inc. Objective 4 Determine the product when given special polynomial factors.

114. Slide 5- 114 Copyright © 2011 Pearson Education, Inc. Multiplying Conjugates If a and b are real numbers, variables, or expressions, then (a + b)(a – b) = a 2 – b 2. Conjugates: Binomials that differ only in the sign separating the terms. x + 9 and x – 9 2x + 3 and 2x – 3 –6x + 5 and –6x – 5

115. Slide 5- 115 Copyright © 2011 Pearson Education, Inc. Example 7 Multiply. a. (x + 5) (x – 5) b. Solution Use (a + b)(a – b) = a 2 – b 2. Simplify. Solution Use (a + b)(a – b) = a 2 – b 2. Simplify.

116. Slide 5- 116 Copyright © 2011 Pearson Education, Inc. Squaring a Binomial If a and b are real numbers, variables, or expressions, then (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2

117. Slide 5- 117 Copyright © 2011 Pearson Education, Inc. Example 8a Multiply. Solution Use (a + b) 2 = a 2 + 2ab + b 2. Simplify.

118. Slide 5- 118 Copyright © 2011 Pearson Education, Inc. Example 8b Multiply. Solution Use (a – b) 2 = a 2 – 2ab + b 2. Simplify.

119. Slide 5- 119 Copyright © 2011 Pearson Education, Inc. Multiply. a) b) c) d) 5.5

120. Slide 5- 120 Copyright © 2011 Pearson Education, Inc. Multiply. a) b) c) d) 5.5

121. Slide 5- 121 Copyright © 2011 Pearson Education, Inc. Multiply. a) b) c) d) 5.5

122. Slide 5- 122 Copyright © 2011 Pearson Education, Inc. Multiply. a) b) c) d) 5.5

123. Copyright © 2011 Pearson Education, Inc. Exponent Rules and Dividing Polynomials 5.6 1.Divide exponential forms with the same base. 2.Divide numbers in scientific notation. 3. Divide monomials. 4.Divide a polynomial by a monomial. 5.Use long division to divide polynomials. 6.Simplify expressions using rules of exponents.

124. Slide 5- 124 Copyright © 2011 Pearson Education, Inc. Objective 1 Divide exponential forms with the same base.

125. Slide 5- 125 Copyright © 2011 Pearson Education, Inc. Quotient Rule for Exponents If m and n are integers and a is a real number, where a 0, then

126. Slide 5- 126 Copyright © 2011 Pearson Education, Inc. Example 1 Divide. Solution Because the exponential forms have the same base, we can subtract the exponents and keep the same base.

127. Slide 5- 127 Copyright © 2011 Pearson Education, Inc. Example 2 Divide and write the result with a positive exponent. Solution Subtract the exponents and keep the same base. Rewrite the subtraction as addition. Simplify. Write with a positive exponent.

128. Slide 5- 128 Copyright © 2011 Pearson Education, Inc. Objective 2 Divide numbers in scientific notation.

129. Slide 5- 129 Copyright © 2011 Pearson Education, Inc. Example 3 Divide and write the result in scientific notation. Solution The decimal factors and powers of 10 can be separated into a product of two fractions, allowing separate division. Note: This is not in scientific notation. We must move the decimal point one place to the right and account for this by decreasing the exponent by one.

130. Slide 5- 130 Copyright © 2011 Pearson Education, Inc. Objective 3 Divide monomials.

131. Slide 5- 131 Copyright © 2011 Pearson Education, Inc. Dividing Monomials To divide monomials, 1. Divide the coefficients or simplify them to fractions in lowest terms. 2. Use the quotient rule for the exponents with like bases. 3.Do not change unlike variable bases in the quotient. 4.Write the final expression so that all exponents are positive.

132. Slide 5- 132 Copyright © 2011 Pearson Education, Inc. Example 4 Divide. Solution

133. Slide 5- 133 Copyright © 2011 Pearson Education, Inc. Objective 4 Divide a polynomial by a monomial.

134. Slide 5- 134 Copyright © 2011 Pearson Education, Inc. If a, b, and c are real numbers, variables, or expressions with c 0, then Dividing a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.

135. Slide 5- 135 Copyright © 2011 Pearson Education, Inc. Example 5 Divide. Solution Divide each term in the polynomial by the monomial.

136. Slide 5- 136 Copyright © 2011 Pearson Education, Inc. Objective 5 Use long division to divide polynomials.

137. Slide 5- 137 Copyright © 2011 Pearson Education, Inc. To divide a polynomial by a polynomial, we can use long division. Divide: Divisor Quotient Remainder Quotient 13 Divisor 12 Remainder 1 = Dividend =157 + +

138. Slide 5- 138 Copyright © 2011 Pearson Education, Inc. Example 6 Divide. Solution Begin by dividing the first term in the dividend by the first term in the divisor: x 2 + 8x + 16. Change signs.

139. Slide 5- 139 Copyright © 2011 Pearson Education, Inc. continued The next term in the quotient is found by dividing the term 5x + 16 by x + 3. Change signs. Answer The next term in the quotient is found by dividing the term 5x + 16 by x + 3.

140. Slide 5- 140 Copyright © 2011 Pearson Education, Inc. Dividing a Polynomial by a Polynomial To divide a polynomial by a polynomial, use long division. If there is a remainder, write the result in the following form:

141. Slide 5- 141 Copyright © 2011 Pearson Education, Inc. Example 7 Divide. Solution Begin by dividing the first term in the dividend by the first term in the divisor: 6b 2 + 5b – 28. Change signs.

142. Slide 5- 142 Copyright © 2011 Pearson Education, Inc. continued Determine the next part of the quotient by dividing 8b by 2b, which is 4, and repeat the multiplication and subtraction steps. Change signs. Answer

143. Slide 5- 143 Copyright © 2011 Pearson Education, Inc. Example 8 Divide. Solution Use a place holder for the missing terms.

144. Slide 5- 144 Copyright © 2011 Pearson Education, Inc. continued Solution Answer Remember to change signs when subtracting.

145. Slide 5- 145 Copyright © 2011 Pearson Education, Inc. Objective 6 Simplify expressions using rules of exponents.

146. Slide 5- 146 Copyright © 2011 Pearson Education, Inc. Exponents Summary Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers. Zero as an exponent:a 0 = 1, where a 0. 0 0 is indeterminate. Negative exponents: Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power:

147. Slide 5- 147 Copyright © 2011 Pearson Education, Inc. Example 9 Simplify. Write all answers with positive exponents. a. Solution a.

148. Slide 5- 148 Copyright © 2011 Pearson Education, Inc. continued b. Solution Use the quotient rule for exponents. Write with a positive exponent.

149. Slide 5- 149 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.6

150. Slide 5- 150 Copyright © 2011 Pearson Education, Inc. Simplify. a) b) c) d) 5.6

151. Slide 5- 151 Copyright © 2011 Pearson Education, Inc. Divide. a) b) c) d) 5.6

152. Slide 5- 152 Copyright © 2011 Pearson Education, Inc. Divide. a) b) c) d) 5.6