46 Chapter P Prerequisites: Fundamental Concepts Algebra Objectives Section Understand the vocabulary polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL in polynomial multiplication. Use special products in polynomial multiplication. Perform operations with polynomials in several variables. P.4 Polynomials Can that be Al, your author s yellow lab, sharing a special moment with a baby chick? And if it is (it is), what possible relevance can this have to polynomials? An answer is promised before you reach the eercise set. For now, we open the section by defining and describing polynomials. How We Define Polynomials More education results in a higher income. The mathematical models M = -18 3 + 923 2-9603 + 48,446 and W = 17 3-450 2 + 6392-14,764 Old Dog Á New Chicks describe the median, or middlemost, annual income for men, M, and women, W, who have completed years education. We ll be working with these models and the data upon which they are based in the eercise set. The algebraic epressions that appear on the right sides the models are eamples polynomials.a polynomial is a single term or the sum two or more containing variables with whole-number eponents. The polynomials above each contain four. Equations containing polynomials are used in such diverse areas as science, business, medicine, psychology, and sociology. In this section, we review basic ideas about polynomials and their operations. Understand the vocabulary polynomials. How We Describe Polynomials Consider the polynomial 7 3-9 2 + 13-6. We can epress this polynomial as 7 3 + 1-9 2 2 + 13 + 1-62. Study Tip We can epress 0 in many ways, including 0, 0 2, and 0 3. It is impossible to assign a unique eponent to the variable. This is why 0 has no defined degree. The polynomial contains four. It is customary to write the in the order descending powers the variable. This is the standard form a polynomial. Some polynomials contain only one variable. Each term such a polynomial in is the form a n. If a Z 0, the degree a n is n. For eample, the degree the term 7 3 is 3. The Degree a n If a Z 0, the degree a n is n. The degree a nonzero constant is 0. The constant 0 has no defined degree.
Section P.4 Polynomials 47 Here is an eample a polynomial and the degree each its four : 6 4-3 3 +2-5. degree 4 degree 3 degree 1 degree nonzero constant: 0 Notice that the eponent on for the term 2, meaning 2 1, is understood to be 1. For this reason, the degree 2 is 1. You can think -5 as -5 0 ; thus, its degree is 0. A polynomial is simplified when it contains no grouping symbols and no like. A simplified polynomial that has eactly one term is called a monomial. A binomial is a simplified polynomial that has two. A trinomial is a simplified polynomial with three. Simplified polynomials with four or more have no special names. The degree a polynomial is the greatest degree all the the polynomial. For eample, 4 2 + 3 is a binomial degree 2 because the degree the first term is 2, and the degree the other term is less than 2. Also, 7 5-2 2 + 4 is a trinomial degree 5 because the degree the first term is 5, and the degrees the other are less than 5. Up to now, we have used to represent the variable in a polynomial. However, any letter can be used. For eample, 7 5-3 3 + 8 is a polynomial (in ) degree 5. Because there are three, the polynomial is a trinomial. 6y 3 + 4y 2 - y + 3 is a polynomial (in y) degree 3. Because there are four, the polynomial has no special name. z 7 + 22 is a polynomial (in z) degree 7. Because there are two, the polynomial is a binomial. We can tie together the threads our discussion with the formal definition a polynomial in one variable. In this definition, the coefficients the are represented by a n (read a sub n ), a n - 1 (read a sub n minus 1 ), a n - 2, and so on. The small letters to the lower right each a are called subscripts and are not eponents. Subscripts are used to distinguish one constant from another when a large and undetermined number such constants are needed. Definition a Polynomial in A polynomial in is an algebraic epression the form a n n + a n - 1 n - 1 + a n - 2 n - 2 + Á + a 1 + a 0, where a n, a n - 1, a n - 2, Á, a 1, and a 0 are real numbers, a n Z 0, and n is a nonnegative integer. The polynomial is degree n, a n is the leading coefficient, and is the constant term. a 0 Add and subtract polynomials. Adding and Subtracting Polynomials Polynomials are added and subtracted by combining like. For eample, we can combine the monomials -9 3 and 13 3 using addition as follows: 9 3 +13 3 =( 9+13) 3 =4 3. These like both contain to the third power. Add coefficients and keep the same variable factor, 3.
48 Chapter P Prerequisites: Fundamental Concepts Algebra EXAMPLE 1 Adding and Subtracting Polynomials Study Tip You can also arrange like in columns and combine vertically: 7 3-8 2 + 9-6 -2 3 + 6 2 + 3-9 5 3-2 2 + 12-15 The like can be combined by adding their coefficients and keeping the same variable factor. Perform the indicated operations and simplify: a. 1-9 3 + 7 2-5 + 32 + 113 3 + 2 2-8 - 62 b. 17 3-8 2 + 9-62 - 12 3-6 2-3 + 92. Solution a. 1-9 3 + 7 2-5 + 32 + 113 3 + 2 2-8 - 62 = 1-9 3 + 13 3 2 + 17 2 + 2 2 2 + 1-5 - 82 + 13-62 Group like. = 4 3 + 9 2 + 1-132 + 1-32 Combine like. = 4 3 + 9 2-13 - 3 Simplify. b. (7 3-8 2 +9-6)-(2 3-6 2-3+9) Change the sign each coefficient. =(7 3-8 2 +9-6)+( 2 3 +6 2 +3-9) Rewrite subtraction as addition the additive inverse. = 17 3-2 3 2 + 1-8 2 + 6 2 2 + 19 + 32 + 1-6 - 92 Group like. = 5 3 + 1-2 2 2 + 12 + 1-152 Combine like. = 5 3-2 2 + 12-15 Simplify. Check Point 1 Perform the indicated operations and simplify: a. 1-17 3 + 4 2-11 - 52 + 116 3-3 2 + 3-152 b. 113 3-9 2-7 + 12-1-7 3 + 2 2-5 + 92. Multiply polynomials. Study Tip Don t confuse adding and multiplying monomials. Addition: Multiplication: Only like can be added or subtracted, but unlike may be multiplied. Addition: 5 4 + 3 2 cannot be simplified. Multiplication: 5 4 + 6 4 = 11 4 15 4 216 4 2 = 15 # 621 4 # 4 2 = 30 4 + 4 = 30 8 15 4 213 2 2 = 15 # 321 4 # 2 2 = 15 4 + 2 = 15 6 Multiplying Polynomials The product two monomials is obtained by using properties eponents. For eample, Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For eample, 3 4 (2 3-7+3)=3 4 2 3-3 4 7+3 4 3=6 7-21 5 +9 4. Monomial Trinomial ( 8 6 )(5 3 )= 8 5 6±3 = 40 9. Multiply coefficients and add eponents. How do we multiply two polynomials if neither is a monomial? For eample, consider (2+3)( 2 +4 +5). Binomial Trinomial
Section P.4 Polynomials 49 One way to perform (2 + 3)( 2 + 4 + 5) is to distribute 2 throughout the trinomial and 3 throughout the trinomial Then combine the like that result. 21 2 + 4 + 52 31 2 + 4 + 52. Multiplying Polynomials When Neither Is a Monomial Multiply each term one polynomial by each term the other polynomial. Then combine like. EXAMPLE 2 Multiplying a Binomial and a Trinomial Multiply: Solution 12 + 321 2 + 4 + 52. 12 + 321 2 + 4 + 52 = 21 2 + 4 + 52 + 31 2 + 4 + 52 = 2 # 2 + 2 # 4 + 2 # 5 + 3 2 + 3 # 4 + 3 # 5 = 2 3 + 8 2 + 10 + 3 2 + 12 + 15 = 2 3 + 11 2 + 22 + 15 Multiply the trinomial by each term the binomial. Use the distributive property. Multiply monomials: Multiply coefficients and add eponents. Combine like : 8 2 + 3 2 = 11 2 and 10 + 12 = 22. Another method for performing the multiplication is to use a vertical format similar to that used for multiplying whole numbers. Write like in the same column. Combine like. 2 +4+ 5 2+ 3 3 2 +12+15 2 3 + 8 2 +10 2 3 +11 2 +22+15 3( 2 + 4 + 5) 2( 2 + 4 + 5) Check Point 2 Multiply: 15-2213 2-5 + 42. Use FOIL in polynomial multiplication. The Two Binomials: FOIL Frequently, we need to find the product two binomials. One way to perform this multiplication is to distribute each term in the first binomial through the second binomial. For eample, we can find the product the binomials 3 + 2 and 4 + 5 as follows: Distribute 3 over 4 + 5. (3+2)(4+5)=3(4+5)+2(4+5) Distribute 2 over 4 + 5. =3(4)+3(5)+2(4)+2(5) =12 2 +15+8+10. We'll combine these like later. For now, our interest is in how to obtain each these four.
50 Chapter P Prerequisites: Fundamental Concepts Algebra We can also find the product 3 + 2 and 4 + 5 using a method called FOIL, which is based on our work shown at the bottom the previous page. Any two binomials can be quickly multiplied by using the FOIL method, in which F represents the product the first in each binomial, O represents the product the outside, I represents the product the inside, and L represents the product the last, or second, in each binomial. For eample, we can use the FOIL method to find the product the binomials 3 + 2 and 4 + 5 as follows: first last F O I L (3+2)(4+5)=12 2 +15+8+10 inside outside First Outside Inside Last =12 2 +23+10 Combine like. In general, here s how to use the FOIL method to find the product and c + d: a + b Using the FOIL Method to Multiply Binomials first last F O I L (a+b)(c+d)=a c+a d+b c+b d inside outside First Outside Inside Last EXAMPLE 3 Using the FOIL Method Multiply: Solution 13 + 4215-32. first last F O I L (3+4)(5-3)=3 5+3( 3)+4 5+4( 3) =15 2-9+20-12 inside outside =15 2 +11-12 Combine like. Check Point 3 Multiply: 17-5214 - 32. Use special products in polynomial multiplication. Special s There are several products that occur so frequently that it s convenient to memorize the form, or pattern, these formulas.
Section P.4 Polynomials 51 Study Tip Although it s convenient to memorize these forms, the FOIL method can be used on all five eamples in the bo. To cube + 4, you can first square + 4 using FOIL and then multiply this result by + 4. In short, you do not necessarily have to utilize these special formulas. What is the advantage knowing and using these forms? Special s Let A and B represent real numbers, variables, or algebraic epressions. Special Sum and Difference Two Terms 1A + B21A - B2 = A 2 - B 2 Squaring a Binomial 1A + B2 2 = A 2 + 2AB + B 2 1A - B2 2 = A 2-2AB + B 2 Cubing a Binomial 1A + B2 3 = A 3 + 3A 2 B + 3AB 2 + B 3 1A - B2 3 = A 3-3A 2 B + 3AB 2 - B 3 Eample 12 + 3212-32 = 122 2-3 2 13-42 2 = 9 2-24 + 16 1 + 42 3 1-22 3 = 4 2-9 1y + 52 2 = y 2 + 2 # y # 5 + 5 2 = y 2 + 10y + 25 = 132 2-2 # 3 # 4 + 4 2 = 3 + 3 2 142 + 3142 2 + 4 3 = 3 + 12 2 + 48 + 64 = 3-3 2 122 + 3122 2-2 3 = 3-6 2 + 12-8 Perform operations with polynomials in several variables. Polynomials in Several Variables A polynomial in two variables, and y, contains the sum one or more monomials in the form a n y m. The constant, a, is the coefficient. The eponents, n and m, represent whole numbers. The degree the monomial a n y m is n + m. Here is an eample a polynomial in two variables: The coefficients are 7, 17, 1, 6, and 9. 7 2 y 3-17 4 y 2 + y - 6y 2 + 9. Degree monomial: 2 + 3 = 5 Degree monomial: 4 + 2 = 6 Degree monomial A1 1 y 1 B: 1 + 1 = 2 Degree monomial A 6 0 y 2 B: 0 + 2 = 2 Degree monomial A9 0 y 0 B: 0 + 0 = 0 The degree a polynomial in two variables is the highest degree all its. For the preceding polynomial, the degree is 6. Polynomials containing two or more variables can be added, subtracted, and multiplied just like polynomials that contain only one variable. For eample, we can add the monomials -7y 2 and 13y 2 as follows: 7y 2 +13y 2 =( 7+13)y 2 =6y 2. These like both contain the variable factors and y 2. Add coefficients and keep the same variable factors, y 2.
52 Chapter P Prerequisites: Fundamental Concepts Algebra EXAMPLE 4 Multiplying Polynomials in Two Variables Multiply: a. 1 + 4y213-5y2 b. 15 + 3y2 2. Solution We will perform the multiplication in part (a) using the FOIL method. We will multiply in part (b) using the formula for the square a binomial sum, 1A + B2 2. a. (+4y)(3-5y) Multiply these binomials using the FOIL method. F O I L =()(3)+()( 5y)+(4y)(3)+(4y)( 5y) =3 2-5y+12y-20y 2 =3 2 +7y-20y 2 Combine like. (A + B) 2 = A 2 + 2 A B + B 2 b. (5+3y) 2 =(5) 2 +2(5)(3y)+(3y) 2 =25 2 +30y+9y 2 Check Point 4 Multiply: a. 17-6y213 - y2 b. 12 + 4y2 2. Special products can sometimes be used to find the products certain trinomials, as illustrated in Eample 5. EXAMPLE 5 Using the Special s Multiply: a. 17 + 5 + 4y217 + 5-4y2 b. 13 + y + 12 2. Solution a. By grouping the first two within each the parentheses, we can find the product using the form for the sum and difference two. (A + B) (A B) = A 2 B 2 [(7+5)+4y] [(7+5)-4y]=(7+5) 2 -(4y) 2 = 172 2 + 2 # 7 # 5 + 5 2-14y2 2 = 49 2 + 70 + 25-16y 2 b. We can group the 13 + y + 12 2 so that the formula for the square a binomial can be applied. (A + B) 2 = A 2 + 2 A B + B 2 [(3+y)+1] 2 =(3+y) 2 +2 (3+y) 1+1 2 = 9 2 + 6y + y 2 + 6 + 2y + 1 Check Point 5 Multiply: a. 13 + 2 + 5y213 + 2-5y2 b. 12 + y + 32 2.
Section P.4 Polynomials 53 Labrador Retrievers and Polynomial Multiplication The color a Labrador retriever is determined by its pair genes. A single gene is inherited at random from each parent. The black-fur gene, B, is dominant. The yellow-fur gene, Y, is recessive. This means that labs with at least one black-fur gene (BB or BY) have black coats. Only labs with two yellow-fur genes (YY) have yellow coats. Al, your author s yellow lab, inherited his genetic makeup from two black BY parents. Second BY parent, a black lab with a recessive yellow-fur gene First BY parent, a black lab with a recessive yellow-fur gene B Y B BB BY Y BY YY The table shows the four possible combinations color genes that BY parents can pass to their fspring. Because YY is one four possible outcomes, the probability that a yellow lab like Al will be the fspring these black parents 1 is 4. The probabilities suggested by the table can be modeled by the epression A 1 2 B + 1 2 YB2. 2 2 1 1 1 a B+ Yb = +2 2 2 a 2 Bb a 1 2 Bb a 1 2 Yb+ 1 2 a 2 Yb = 1 4 BB + 1 2 BY + 1 4 YY The probability a black lab with two dominant black genes is 1. 4 The probability a black lab with a recessive yellow gene is 1. 2 The probability a yellow lab with two recessive yellow genes is 4 1. Eercise Set P.4 Practice Eercises In Eercises 1 4, is the algebraic epression a polynomial? If it is, write the polynomial in standard form. 1. 2 + 3 2-5 2. 2 + 3-1 - 5 3. 2 + 3 4. 2-3 + 4-5 In Eercises 5 8, find the degree the polynomial. 5. 3 2-5 + 4 6. -4 3 + 7 2-11 7. 2-4 3 + 9-12 4 + 63 8. 2-8 3 + 15 4 + 91 In Eercises 9 14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. 9. 1-6 3 + 5 2-8 + 92 + 117 3 + 2 2-4 - 132 10. 1-7 3 + 6 2-11 + 132 + 119 3-11 2 + 7-172 11. 117 3-5 2 + 4-32 - 15 3-9 2-8 + 112 12. 118 4-2 3-7 + 82-19 4-6 3-5 + 72 13. 15 2-7 - 82 + 12 2-3 + 72-1 2-4 - 32 14. 18 2 + 7-52 - 13 2-42 - 1-6 3-5 2 + 32 In Eercises 15 82, find each product. 15. 1 + 121 2 - + 12 16. 1 + 521 2-5 + 252 17. 12-321 2-3 + 52 18. 12-121 2-4 + 32 19. 1 + 721 + 32 20. 1 + 821 + 52 21. 1-521 + 32 22. 1-121 + 22 23. 13 + 5212 + 12 24. 17 + 4213 + 12 25. 12-3215 + 32 26. 12-5217 + 22 27. 15 2-4213 2-72 28. 17 2-2213 2-52 29. 18 3 + 321 2-52 30. 17 3 + 521 2-22 31. 1 + 321-32 32. 1 + 521-52 33. 13 + 2213-22 34. 12 + 5212-52 35. 15-7215 + 72 36. 14-3214 + 32 37. 14 2 + 5214 2-52 38. 13 2 + 4213 2-42 39. 11 - y 5 211 + y 5 2 40. 41. 1 + 22 2 42. 1 + 52 2 12 - y 5 212 + y 5 2
54 Chapter P Prerequisites: Fundamental Concepts Algebra 43. 12 + 32 2 44. 45. 1-32 2 46. 47. 14 2-12 2 48. 49. 17-22 2 50. 51. 1 + 12 3 52. 53. 12 + 32 3 54. 55. 1-32 3 56. 57. 13-42 3 58. 59. 1 + 5y217 + 3y2 60. 61. 1-3y212 + 7y2 62. 63. 13y - 1215y + 22 64. 65. 17 + 5y2 2 66. 67. 1 2 y 2-32 2 68. 69. 1 - y21 2 + y + y 2 2 70. 71. 13 + 5y213-5y2 72. 13 + 22 2 1-42 2 15 2-32 2 19-52 2 1 + 22 3 13 + 42 3 1-12 3 12-32 3 1 + 9y216 + 7y2 13 - y212 + 5y2 17 2 y + 1212 2 y - 32 19 + 7y2 2 1 2 y 2-52 2 1 + y21 2 - y + y 2 2 17 + 3y217-3y2 Application Eercises As you complete more years education, you can count on a greater income. The bar graph shows the median, or middlemost, annual income for Americans, by level education, in 2004. Median Annual Income (thousands dollars) $90 $80 $70 $60 $50 $40 $30 $20 $10 Median Annual Income, by Level Education, 2004 21,659 17,023 8 26,277 19,169 10 Source: Bureau the Census 35,725 26,029 Men 41,895 30,816 44,404 33,481 12 13 14 16 Years School Completed Here are polynomial models that describe the median annual income for men, M, and for women, W, who have completed years education: M = 177 2 + 288 + 7075 W = 255 2-2956 + 24,336 Women 57,220 41,681 71,530 51,316 18 82,401 68,875 20 73. 74. 1 + y + 321 + y - 32 1 + y + 521 + y - 52 M = -18 3 + 923 2-9603 + 48,446 W = 17 3-450 2 + 6392-14,764. 75. 13 + 7-5y213 + 7 + 5y2 76. 15 + 7y - 2215 + 7y + 22 77. 35y - 12 + 32435y + 12 + 324 78. 38y + 17-32438y - 17-324 79. 1 + y + 12 2 80. 1 + y + 22 2 81. 12 + y + 12 2 82. 15 + 1 + 6y2 2 Practice Plus In Eercises 83 90, perform the indicated operation or operations. 83. 13 + 4y2 2-13 - 4y2 2 84. 15 + 2y2 2-15 - 2y2 2 85. 15-7213 - 22-14 - 5216-12 86. 13 + 5212-92 - 17-221 - 12 87. 12 + 5212-5214 2 + 252 88. 13 + 4213-4219 2 + 162 12-72 5 15-32 6 89. 90. 12-72 3 15-32 4 Eercises 91 92 are based on these models and the data displayed by the graph. 91. a. Use the equation defined by a polynomial degree 2 to find the median annual income for a man with 16 years education. Does this underestimate or overestimate the median income shown by the bar graph? By how much? b. Use the equations defined by polynomials degree 3 to find a mathematical model for M - W. c. According to the model in part (b), what is the difference in the median annual income between men and women with 14 years education? d. According to the data displayed by the graph, what is the actual difference in the median annual income between men and women with 14 years education? Did the result part (c) underestimate or overestimate this difference? By how much? 92. a. Use the equation defined by a polynomial degree 2 to find the median annual income for a woman with 18 years education. Does this underestimate or overestimate the median income shown by the bar graph? By how much? b. Use the equations defined by polynomials degree 3 to find a mathematical model for M - W. c. According to the model in part (b), what is the difference in the median annual income between men and women with 16 years education? d. According to the data displayed by the graph, what is the actual difference in the median annual income between men and women with 16 years education? Did the result part (c) underestimate or overestimate this difference? By how much?
Section P.4 Polynomials 55 The volume, V, a rectangular solid with length l, width w, and 104. I used the FOIL method to find the product + 5 and height h is given by the formula V = lwh. In Eercises 93 94, use 2 + 2 + 1. this formula to write a polynomial in standard form that models, 105. Many English words have prefies with meanings similar to or represents, the volume the open bo. those used to describe polynomials, such as monologue, 93. binocular, and tricuspid. 94. 8 2 8 2 5 2 10 2 106. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method. 107. Epress the area the plane figure shown as a polynomial in standard form. 1 In Eercises 95 96, write a polynomial in standard form that models, or represents, the area the shaded region. 95. + 9 + 3 + 1 + 5 In Eercises 108 109, represent the volume each figure as a polynomial in standard form. 108. + 3 96. + 4 + 2 2 1 + 1 + 3 + 1 + 3 Writing in Mathematics 97. What is a polynomial in? 98. Eplain how to subtract polynomials. 99. Eplain how to multiply two binomials using the FOIL method. Give an eample with your eplanation. 100. Eplain how to find the product the sum and difference two. Give an eample with your eplanation. 101. Eplain how to square a binomial difference. Give an eample with your eplanation. 102. Eplain how to find the degree a polynomial in two variables. Critical Thinking Eercises Make Sense? In Eercises 103 106, determine whether each statement makes sense or does not make sense, and eplain your reasoning. 103. Knowing the difference between factors and is important: In 13 2 y2 2, I can distribute the eponent 2 on each factor, but in 13 2 + y2 2, I cannot do the same thing on each term. 109. 110. Simplify: 1y n + 221y n - 22-1y n - 32 2. Preview Eercises Eercises 111 113 will help you prepare for the material covered in the net section. In each eercise, replace the boed question mark with an integer that results in the given product. Some trial and error may be necessary. 111. 1 + 321 +? 2 = 2 + 7 + 12 112. + 5 3 2 + 1 + 2 1 -? 21-122 = 2-14 + 24 113. 14 + 1212 -? 2 = 8 2-10 - 3