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38 Chapter P P.4 Prerequisites Operations with Polynomials What you should learn: • Write polynomials in standard form and identify the leading coefficients and degrees of polynomials • Add and subtract polynomials • Multiply polynomials • Use special products to multiply polynomials • Use operations with polynomials in application problems Why you should learn it: Operations with polynomials enable you to model various aspects of the physical world, such as the position of a free-falling object, as shown in Exercises 163–168 on page 50. Basic Definitions An algebraic expression containing only terms of the form axk, where a is any real number and k is a nonnegative integer, is called a polynomial in one variable or simply a polynomial. Here are some examples of polynomials in one variable. 3x 8, x4 3x3 x2 8x 1, x3 5, and 9x5 In the term axk, a is called the coefficient, and k the degree, of the term. Note that the degree of the term ax is 1, and the degree of a constant term is 0. Because a polynomial is an algebraic sum, the coefficients take on the signs between the terms. For instance, x3 4x2 3 1x3 4x2 0x 3 has coefficients 1, 4, 0, and 3. Polynomials are usually written in order of descending powers of the variable. This is referred to as standard form. For example, the standard form of 3x2 5 x3 2x is x3 3x2 2x 5. Standard form The degree of a polynomial is defined as the degree of the term with the highest power, and the coefficient of this term is called the leading coefficient of the polynomial. For instance, the polynomial 3x4 4x2 x 7 is of fourth degree and its leading coefficient is 3. Definition of Polynomial in x Let a0, a1, a2, a3, . . . , an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form an x n an1x n1 . . . a2 x2 a1x a0 where an 0. The polynomial is of degree n, and the number an is called the leading coefficient. The number a0 is called the constant term. The following are not polynomials, for the reasons stated. • The expression 2x1 5 is not a polynomial because the exponent in 2x1 is negative. • The expression x3 3x12 is not a polynomial because the exponent in 3x12 is not a nonnegative integer. Section P.4 Operations with Polynomials 39 Example 1 Identifying Leading Coefficients and Degrees Write the polynomial in standard form and identify the degree and leading coefficient of the polynomial. (a) 5x2 2x7 4 2x (b) 16 8x3 (c) 5 x 4 6x3 Solution Standard Form Degree Leading Coefficient (a) 5x2 2x7 4 2x 2x7 5x2 2x 4 7 2 (b) 16 8x3 3 8 4 1 Polynomial 8x3 (c) 5 x 4 6x3 16 x 4 6x3 5 Now try Exercise 7. A polynomial with only one term is a monomial. Polynomials with two unlike terms are binomials, and those with three unlike terms are trinomials. Here are some examples. Monomial: 5x3 Binomial: 4x 3 Trinomial: 2x2 3x 7 The prefix mono means one, the prefix bi means two, and the prefix tri means three. Example 2 Evaluating a Polynomial Find the value of x3 5x2 6x 3 when x 4. Solution When x 4, the value of x3 5x2 6x 3 is x3 5x2 6x 3 43 542 64 3 Substitute 4 for x. 64 80 24 3 Evaluate terms. 5 Simplify. Now try Exercise 27. Adding and Subtracting Polynomials To add two polynomials, simply combine like terms. This can be done in either a horizontal or a vertical format, as shown in Examples 3 and 4. Example 3 Adding Polynomials Horizontally Use a horizontal format to add 2x3 x2 5 and x2 x 6. Solution 2x3 x2 5 x2 x 6 2x3 x2 x2 x 5 6 2x3 2x2 x1 Now try Exercise 31. Write original polynomials. Group like terms. Combine like terms. 40 Chapter P Prerequisites To use a vertical format to add polynomials, align the terms of the polynomials by their degrees, as shown in the following example. Example 4 Using a Vertical Format to Add Polynomials Use a vertical format to add 5x3 2x2 x 7, 3x2 4x 7, and x3 4x2 8. Solution 5x3 2x2 x 7 3x2 4x 7 x3 4x3 4x2 9x2 8 5x 6 Now try Exercise 33. To subtract one polynomial from another, add the opposite. You can do this by changing the sign of each term of the polynomial that is being subtracted and then adding the resulting like terms. Example 5 Subtracting Polynomials Horizontally Use a horizontal format to subtract x3 2x2 x 4 from 3x3 5x2 3. Solution Write original polynomials. 3x3 5x2 3 x3 2x2 x 4 3x3 5x2 3 x3 2x2 x 4 3x3 x3 5x2 2x2 x 3 4 2x3 7x2 x 7 Add the opposite. Group like terms. Combine like terms. Now try Exercise 39. Study Tip The common error illustrated to the right is forgetting to change two of the signs in the polynomial that is being subtracted. When subtracting polynomials, remember to add the opposite of every term of the subtracted polynomial. Be especially careful to get the correct signs when you are subtracting one polynomial from another. One of the most common mistakes in algebra is to forget to change signs correctly when subtracting one expression from another. Here is an example. Wrong sign x2 2x 3 x2 2x 2 x2 2x 3 x2 2x 2 Common error Wrong sign Example 6 Using a Vertical Format to Subtract Polynomials Use a vertical format to subtract 3x4 2x3 3x 4 from 4x4 2x3 5x2 x 8. Solution 4x4 2x3 5x2 x 8 3x4 2x3 3x 4 4x4 2x3 5x2 x 8 3x4 2x3 x4 Now try Exercise 45. 3x 4 5x2 4x 12 Section P.4 Operations with Polynomials 41 Multiplying Polynomials The simplest type of polynomial multiplication involves a monomial multiplier. The product is obtained by direct application of the Distributive Property. For instance, to multiply the monomial 3x by the polynomial 2x2 5x 3, multiply each term of the polynomial by 3x. 3x2x2 5x 3 3x2x2 3x5x 3x3 6x3 15x2 9x Example 7 Finding Products with Monomial Multipliers Multiply the polynomial by the monomial. (a) 2x 73x (b) 4x22x3 3x 1 Solution (a) 2x 73x 2x3x 73x 6x2 Distributive Property 21x Properties of exponents (b) 4x22x3 3x 1 4x22x3 4x23x 4x21 8x5 12x3 4x2 Distributive Property Properties of exponents Now try Exercise 71. To multiply two binomials, you can use both (left and right) forms of the Distributive Property. For example, if you treat the binomial 2x 7 as a single quantity, you can multiply 3x 2 by 2x 7 as follows. 3x 22x 7 3x2x 7 22x 7 3x2x 3x7 22x 27 6x2 21x 4x 14 Outer First Product of First terms 3x 22x 7 Inner Last FOIL Diagram Product of Outer terms Product of Inner terms Product of Last terms 6x2 17x 14 The four products in the boxes above suggest that you can put the product of two binomials in the FOIL form in just one step. This is called the FOIL Method. Note that the words first, outer, inner, and last refer to the positions of the terms in the original product (see diagram at the left). Example 8 Multiplying Binomials (Distributive Property) Use the Distributive Property to multiply x 2 by x 3. Solution x 2x 3 xx 3 2x 3 x2 3x 2x 6 x2 x 6 Now try Exercise 77. Distributive Property Distributive Property Combine like terms. 42 Chapter P Prerequisites Example 9 Multiplying Binomials (FOIL Method) Use the FOIL method to multiply the binomials. (a) x 3x 9 (b) 3x 42x 1 Solution F O I L (a) x 3x 9 x2 9x 3x 27 x2 12x 27 F O I L (b) 3x 42x 1 6x2 3x 8x 4 6x2 11x 4 Now try Exercise 81. To multiply two polynomials that have three or more terms, you can use the same basic principle that you use when multiplying monomials and binomials. That is, each term of one polynomial must be multiplied by each term of the other polynomial. This can be done using either a horizontal or a vertical format. Example 10 Multiplying Polynomials (Horizontal Format) 4x2 3x 12x 5 4x22x 5 3x2x 5 12x 5 8x3 20x2 6x2 15x 2x 5 8x3 26x2 13x 5 Distributive Property Distributive Property Combine like terms. Now try Exercise 97. When multiplying two polynomials, it is best to write each in standard form before using either the horizontal or vertical format. This is illustrated in the next example. Example 11 Multiplying Polynomials (Vertical Format) Write the polynomials in standard form and use a vertical format to multiply. 4x2 x 25 3x x2 Solution With a vertical format, line up like terms in the same vertical columns, much as you align digits in whole-number multiplication. 4x2 x2 4 3 4x x 2x2 12x3 3x2 20x2 4x 4 11x3 25x2 x 2 3x 5 Write in standard form. Write in standard form. x24x2 x 2 6x 5x 10 x 10 Now try Exercise 101. 3x4x2 x 2 54x2 x 2 Section P.4 Operations with Polynomials 43 Polynomials are often written with exponents. As shown in the next example, the properties of algebra are used to simplify these expressions. EXPLORATION Use the FOIL Method to find the product of x ax a Example 12 Multiplying Polynomials Expand x 43. where a is a constant. What do you notice about the number of terms in your product? What degree are the terms in your product? Solution x 43 x 4x 4x 4 x 4x 4x 4 Write each factor. Associative Property of Multiplication x2 4x 4x 16x 4 Multiply x 4x 4. Combine like terms. x2 8x 16x 4 x2x 4 8xx 4 16x 4 Distributive Property Distributive Property x3 4x2 x3 12x2 8x2 32x 16x 64 48x 64 Combine like terms. Now try Exercise 129. Example 13 An Area Model for Multiplying Polynomials Show that x 22x 1 2x2 5x 2. Solution An appropriate area model to demonstrate the multiplication of two binomials would be A lw, the area formula for a rectangle. Think of a rectangle whose sides are x 2 and 2x 1. The area of this rectangle is x 22x 1. x x 1 1 x+2 x Another way to find the area is to add the areas of the rectangular parts, as shown in Figure P.11. There are two squares whose sides are x, five rectangles whose sides are x and 1, and two squares whose sides are 1. The total area of these nine rectangles is 2x2 5x 2. 2x + 1 Figure P.11 Area widthlength Area sum of rectangular areas Because each method must produce the same area, you can conclude that x 22x 1 2x2 5x 2. Now try Exercise 155. Special Products Some binomial products have special forms that occur frequently in algebra. For instance, the product x 3x 3 is called the product of the sum and difference of two terms. With such products, the two middle terms subtract out, as follows. x 3x 3 x2 3x 3x 9 x2 9 Sum and difference of two terms Product has no middle term. 44 Chapter P Prerequisites Another common type of product is the square of a binomial. With this type of product, the middle term is always twice the product of the terms in the binomial. 2x 52 2x 52x 5 4x2 10x 10x 25 4x2 20x 25 Square of a binomial Outer and inner terms are equal. Middle term is twice the product of the terms in the binomial. Special Products Let u and v be real numbers, variables, or algebraic expressions. Then the following formulas are true. Sum and Difference of Same Terms Example u vu v u v 3x 43x 4 3x2 42 9x2 16 Square of a Binomial Example 2 u v 2 u2 2uv 2 4x 92 4x2 24x9 92 16x2 72x 81 v2 u v2 u2 2uv v2 a a b a2 ab x 62 x2 2x6 62 x2 12x 36 The square of a binomial can also be demonstrated geometrically. Consider a square, each of whose sides are of length a b. (See Figure P.12). The total area includes one square of area a2, two rectangles of area ab each, and one square of area b2. So, the total area is a2 2ab b2. a+b Example 14 Finding Special Products b b2 ab a+b Figure P.12 Multiply the polynomials. (a) 3x 23x 2 (b) 2x 72 (c) a 2 b2 Solution (a) 3x 23x 2 3x2 22 Sum and difference of same terms 9x2 4 (b) 2x 7 2x 22x7 2 2 Simplify. 72 Square of a binomial 4x2 28x 49 Simplify. (c) a 2 b a 2 2a 2b b 2 2 2 a2 4a 4 2ab 4b b2 Now try Exercise 107. Square of a binomial Simplify. Section P.4 Operations with Polynomials 45 Applications There are many applications that require the evaluation of polynomials. One commonly used second-degree polynomial is called a position polynomial. This polynomial has the form 16t2 v0 t s0 Position polynomial where t is the time, measured in seconds. The value of this polynomial gives the height (in feet) of a free-falling object above the ground, assuming no air resistance. The coefficient of t, v0, is called the initial velocity of the object, and the constant term, s0, is called the initial height of the object. If the initial velocity is positive, the object was projected upward (at t 0), if the initial velocity is negative, the object was projected downward, and if the initial velocity is zero, the object was dropped. Example 15 Finding the Height of a Free-Falling Object t=0 t=1 An object is thrown downward from the top of a 200-foot building. The initial velocity is 10 feet per second. Use the position polynomial 16t2 10t 200 to find the height of the object when t 1, t 2, and t 3 (see Figure P.13). 200 ft t=2 Solution When t 1, the height of the object is Height 1612 101 200 t=3 16 10 200 174 feet. Figure P.13 When t 2, the height of the object is Height 1622 102 200 64 20 200 116 feet. When t 3, the height of the object is Height 1632 103 200 144 30 200 26 feet. Now try Exercise 167. In Example 15, the initial velocity is 10 feet per second. The value is negative because the object was thrown downward. If it had been thrown upward, the initial velocity would have been positive. If it had been dropped, the initial velocity would have been zero. Use your calculator to determine the height of the object in Example 15 when t 3.2368. What can you conclude? 46 Chapter P Prerequisites Example 16 Using Polynomial Models The numbers of vehicles (in thousands) fueled by compressed natural gas G and by electricity E in the United States from 1995 to 2003 can be modeled by G 0.079t2 8.95t 3.2, 5 ≤ t ≤ 13 E 1.090t2 14.73t 51.6, 5 ≤ t ≤ 13 Vehicles fueled by natural gas Vehicles fueled by electricity where t represents the year, with t 5 corresponding to 1995. Find a model that represents the total numbers T of vehicles fueled by compressed natural gas and by electricity from 1995 to 2003. Then estimate the total number T of vehicles fueled by compressed natural gas and by electricity in 2002. (Source: Science Applications International Corporation and Energy Information Administration) Solution The sum of the two polynomial models is as follows. G E 0.079t2 8.95t 3.2 1.090t2 14.73t 51.6 1.169t2 5.78t 54.8 So, the polynomial that models the total numbers of vehicles fueled by compressed natural gas and by electricity is TGE 1.169t2 5.78t 54.8 Using this model, and substituting t 12, you can estimate the total number of vehicles fueled by compressed natural gas and by electricity in 2002 to be T 1.169122 5.7812 54.8 153.776 thousand vehicles. Now try Exercise 169. Example 17 Geometry: Finding the Area of a Shaded Region Find an expression for the area of the shaded portion in Figure P.14. 2x + 5 x+3 x−3 x+1 Solution First find the area of the large rectangle A1 and the area of the small rectangle A2. A1 2x 5x 1 and Figure P.14 A2 x 3x 3 Then to find the area A of the shaded portion, subtract A2 from A1. A A1 A2 Write formula. 2x 5x 1 x 3x 3 Substitute. 2x2 7x 5 x2 9 Use FOIL Method and special product formula. 2x2 7x 5 x2 9 Distributive Property. x2 7x 14 Combine like terms. Now try Exercise 149. Section P.4 P.4 Operations with Polynomials 47 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The expression an x n an1 x n1 . . . a2 x2 a1x a0an 0 is called a ________. 2. The ________ of a polynomial is the degree of the term with the highest power, and the coefficient of this term is the ________ of the polynomial. 3. A polynomial with one term is called a ________, while a polynomial with two unlike terms is called a ________, and a polynomial with three unlike terms is called a ________. 4. The letters in “FOIL” stand for the following. F ________ O ________ I ________ L ________ 5. The product u vu v u2 v2 is called the ________ and ________ of ________ terms. 6. The product u v2 u2 2uv v2 is called the ________ of a ________. 7. The expression 16t2 v0t s0 is called the ________, and v0 is the initial ________ and s0 is the initial ________. In Exercises 1–12, write the polynomial in standard form, and find its degree and leading coefficient. 1. 10x 4 2. 3. 5 4. 3x3 2x2 3 3y4 3x2 8 5. 8z 16z2 6. 35t 16t2 7. 6t 4t5 t2 3 8. 10 3x2 15x5 7x 9. x 5 x3 5x2 11. x 10. 16 z2 8z 4z3 12. 4 29. x4 4x3 16x 16 (a) x 1 (b) x 52 30. 3t 4 4t3 (a) t 1 (b) t 23 In Exercises 31–34, perform the addition using a horizontal format. 31. 2x2 3 5x2 6 32. 3x3 2x 8 3x 5 33. x2 3x 8 2x2 4x 3x2 34. 5y 6 4y2 6y 3 9 2y 11y2 In Exercises 13–18, determine whether the polynomial is a monomial, binomial, or trinomial. 13. 12 5y2 14. t3 15. x3 2x2 4 16. 2u7 9u3 17. 1.3x2 18. 2 x4 4z2 In Exercises 35–38, perform the addition using a vertical format. 35. 5x2 3x 4 3x2 4 In Exercises 19–26, give an example of a polynomial in one variable satisfying the conditions. (There are many correct answers.) 19. A monomial of degree 3 20. A trinomial of degree 3 36. 4x3 2x2 8x 4x2 x 6 37. 2b 3 b2 2b 7 b2 38. v2 v 3 4v 1 2v2 3v In Exercises 39–42, perform the subtraction using a horizontal format. 39. 3x2 2x 1 2x2 x 1 21. A trinomial of degree 4 and leading coefficient 2 22. A binomial of degree 2 and leading coefficient 8 23. A monomial of degree 1 and leading coefficient 7 24. A binomial of degree 5 and leading coefficient 3 40. 5y4 2 3y4 2 41. 10x3 15 6x3 x 11 42. y2 3y4 y4 y2 25. A monomial of degree 0 In Exercises 43–46, perform the subtraction using a vertical format. 26. A monomial of degree 2 and leading coefficient 9 43. x2 x 3 x 2 In Exercises 27–30, evaluate the polynomial for each specified value of the variable. 27. x3 12x (a) x 2 (b) x 0 28. 14x4 2x2 (a) x 2 (b) x 2 44. 3z2 z z3 2z2 z 45. 2x3 15x 25 2x3 13x 12 46. 0.2t 4 5t 2 t 4 0.3t 2 1.4 48 Chapter P Prerequisites In Exercises 47–68, perform the indicated operation(s). 91. 3x5x5x 2 47. 92. 4t3tt2 1 2 48. 20s 12s 32 15s2 6s 2 2 49. 4x 5x 6 2x 4x 5 3 2 3 50. 13x 9x 4x 5 5x 7x 3 51. 10x2 11 7x3 12x2 15 52. 15y4 18y 18 11y4 8y 8 53. 5s 6s 30s 8 54. 3x2 23x 9 x2 55. 8x3 4x2 3x x3 4x2 5 x 5 56. 5y2 2y y2 y 3y2 6y 2 57. 52x3 1 3x3 12x2 4x 2 3x2 2x 1 58. 2 y2 3y 9 34y 4 5y2 2y 3 59. 2t2 12 5t2 5 6t2 5 60. 10v 2 8v 1 3v 9 61. 2z2 z 11 3z2 4z 5 22z2 5z 10 62. 73t4 2t2 t 5t4 9t2 4t 38t2 5t 63. 25x3 13x 4x3 9x2 3x 3x3 2x2 6x 5 64. 5t3 2t2 t 8 3t3 t2 4t 2 42t2 3t 1 t3 1 2 2 65. 8.04x 9.37x 5.62x2 66. 11.98y3 4.63y3 6.79y3 67. 4.098a2 6.349a 11.246a2 9.342a 68. 27.433k2 19.018k 14.61k2 3.814k 3x2 8 7 5x2 In Exercises 69–96, perform the multiplication and simplify. 69. 2a28a 70. 6n3n2 71. 2y5 y 72. 5z2z 7 73. 4x32x2 3x 5 74. 3y23y2 7y 3 75. 2x25 3x2 7x3 76. 3a211a 3 77. x 7x 4 78. y 2 y 3 79. 5 x3 x 80. 2 y4 y 81. 2t 1t 8 82. 3z 52z 7 83. 3a4 52a4 7 84. 8b5 13b5 2 85. 2x y3x 2y 86. 2x y3x 2y 87. 5x2 3yx2 y 88. a3 2b54a3 3b5 89. 4y 13 12y 9 90. 5t 34 2t 16 93. 5aa 2 3a2a 3 94. 4x2x 1 9xx 3 95. 2t 1t 1 32t 5 96. 58y 3 2y 1y 7 In Exercises 97–100, perform the multiplication using a horizontal format. 97. x3 3x 2x 2 98. t 3t2 5t 1 99. u 52u2 3u 4 100. x 1x2 4x 6 In Exercises 101–104, perform the multiplication using a vertical format. 101. 7x2 14x 9x 3 102. 4x4 6x2 92x2 3 103. x2 2x 12x 1 104. 2s2 5s 63s 4 In Exercises 105–138, perform the multiplication. 105. x 4x 4 106. y 7y 7 107. a 6ca 6c 108. 8n m8n m 109. 2t 92t 9 110. 5z 15z 1 111. 2x 1 4 2x 1 4 112. 23 x 723 x 7 113. 0.2t 0.50.2t 0.5 114. 4a 0.1b4a 0.1b 115. x3 4x3 4 116. a5 3a5 3 117. x 52 118. x 92 119. 5x 2 120. 3x 82 121. 2a 3b2 122. 4x 5y2 123. 2x4 32 124. y7 4z2 2 125. x2 4x2 2x 4 126. 2x2 32x2 2x 3 127. t2 5t 12t2 5 128. 2z2 3z 73z 4 129. a 53 130. y 23 131. 2x 33 132. 3y 43 133. a2 9a 5a2 a 3 134. t2 2t 72t2 8t 3 135. x 2 y2 136. x 4 y2 137. 2z y 12 138. u v 32 Section P.4 In Exercises 139–142, perform the indicated operations and simplify. 139. x 3x 3 x2 8x 2 Operations with Polynomials 49 Geometric Modeling In Exercises 153–156, (a) perform the multiplication algebraically and (b) use a geometric area model to verify your solution to part (a). 153. xx 3 140. k 8k 8 k2 k 3 154. 2yy 1 141. t 32 t 32 155. t 3t 2 142. a 62 a 62 156. 2z 5z 1 Geometry In Exercises 143–146, write an expression for the perimeter or circumference of the figure. 143. a 144. x2 x x a+3 a+3 Geometric Modeling In Exercises 157 and 158, use the area model to write two different expressions for the total area. Then equate the two expressions and name the algebraic property that is illustrated. 157. x x2 a x a x+b 145. 146. 3y 2y b 2x − 4 y+4 x+a x 158. 4y − 5 a x Geometry In Exercises 147–152, write an expression for the area of the shaded portion of the figure. 147. 148. 3x + 1 x+a a 2x − 5 y+2 x+a 149. 159. Geometry The length of a rectangle is 112 times its width w. Write expressions for (a) the perimeter and (b) the area of the rectangle. 3x + 10 160. Geometry The base of a triangle is 3x and its height is x 5. Write an expression for the area A of the triangle. x 3x x +4 150. 151. 4x + 2 5t 2 x−1 x−1 4x + 2 6t 7t 4 152. 1.6x 0.8x x 2x 6t 161. Compound Interest After 2 years, an investment of $1000 compounded annually at an interest rate of r will yield an amount 10001 r2. Find this product. 162. Compound Interest After 2 years, an investment of $1000 compounded annually at an interest rate of 3.5% will yield an amount 10001 0.0352. Find this product. Chapter P Prerequisites Free-Falling Object In Exercises 163–166, use the position polynomial to determine whether the free-falling object was dropped, thrown upward, or thrown downward. Then determine the height of the object at time t 0. 163. 16t2 100 164. 16t2 50t 165. 16t2 24t 50 166. 16t2 32t 300 167. Free-Falling Object An object is thrown upward from the top of a 200-foot building (see figure). The initial velocity is 40 feet per second. Use the position polynomial 16t2 40t 200 to find the height of the object when t 1, t 2, and t 3. (b) During the given time period, the per capita consumption of beverage milks was decreasing and the per capita consumption of bottled water was increasing (see figure). Was the combined per capita consumption of both beverage milks and bottled water increasing or decreasing over the given time period? y Per capita consumption (in gallons) 50 30 Beverage milks Bottled water 25 20 15 10 t 5 200 ft 250 ft 6 7 8 9 10 11 12 13 Year (5 ↔1995) Figure for 169 Synthesis Figure for 167 Figure for 168 168. Free-Falling Object An object is thrown downward from the top of a 250-foot building (see figure). The initial velocity is 25 feet per second. Use the position polynomial 16t2 25t 250 to find the height of the object when t 1, t 2, and t 3. 169. Beverage Consumption The per capita consumption of all beverage milks M and bottled water W in the United States from 1995 to 2003 can be approximated by the following two polynomial models. M 0.0009t2 0.288t 25.44 Beverage milks W 0.0544t2 0.268t 9.40 Bottled water In these models, the per capita consumption is given in gallons and t5 ≤ t ≤ 13 represents the year, with t 5 corresponding to 1995. (Source: USDA/Economic Research Service) (a) Find a polynomial model that represents the per capita consumption of both beverage milks and bottled water during the given time period. Use this model to find the per capita consumption of beverage milks and bottled water in 1999 and 2003. 170. Writing Explain why x y2 is not equal to x2 y2. 171. Think About It Is every trinomial a seconddegree polynomial? Explain. 172. Think About It Can two third-degree polynomials be added to produce a second-degree polynomial? If so, give an example. 173. Perform the multiplications. (a) x 1x 1 (b) x 1x2 x 1 (c) x 1x3 x2 x 1 From the pattern formed by these products, can you predict the result of x 1x4 x3 x2 x 1? 174. Writing Explain why x2 3x23 is not a polynomial.