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6.4 Multiplication of Polynomials and Special Products 6.4 OBJECTIVES 1. 2. 3. 4. Evaluate f(x) g(x) for a given x Multiply two polynomial functions Square a binomial Find the product of two binomials as a difference of squares In Section 1.4, you saw the first exponent property and used that property to multiply monomials. Let’s review. Example 1 Multiplying Monomials Multiply. Add exponents. (8x2y)(4x3y4) (8 4)(x23)(y14) NOTE a a a m n mn Multiply. Notice the use of the associative and commutative properties to “regroup” and “reorder” the factors. 32x y 5 5 CHECK YOURSELF 1 Multiply. (a) (4a3b)(9a3b2) (b) (5m3n)(7mn5) We now want to extend the process to multiplying polynomial functions. Example 2 Multiplying a Monomial and a Binomial Function Given f(x) 5x2 and g(x) 3x2 5x, and letting h(x) f(x) g(x), find h(x). h(x) f(x) g(x) 5x2 (3x2 5x) © 2001 McGraw-Hill Companies Apply the distributive property. 5x 3x 5x 5x 2 2 2 15x4 25x3 CHECK YOURSELF 2 Given f(x) 3x2 and g(x) 4x2 x, and letting h(x) f(x) g(x), find h(x). We can check this result by comparing the values of h(x) and of f(x) g(x) for a specific value of x. This is illustrated in Example 3. 403 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Example 3 Multiplying a Monomial and a Binomial Function Given f(x) 5x2 and g(x) 3x25x, and letting h(x) f(x) g(x), compare f(1) g(1) with h(1). f(1) g(1) 5(1)2 (3(1)2 5(1)) 5(3 5) 5(2) 10 From Example 2, we know that h(x) 15x4 25x3 So h(1) 15(1)4 25(1)3 15 25 10 Therefore, h(1) f(1) g(1). CHECK YOURSELF 3 Given f(x) 3x2 and g(x) 4x2 x, and letting h(x) f(x) g(x), compare f (1) g(1) with h(1). The distributive property is also used to multiply two polynomial functions. To consider the pattern, let’s start with the product of two binomial functions. Example 4 Multiplying Binomial Functions Given f(x) x 3 and g(x) 2x 5, and letting h(x) f(x) g(x), find the following. (a) h(x) h(x) f(x) g(x) (x 3)(2x 5) Apply the distributive property. (x 3)(2x) (x 3)(5) Apply the distributive property again. (x)(2x) (3)(2x) (x)(5) (3)(5) 2x2 6x 5x 15 2x2 11x 15 Notice that this ensures that each term in the first polynomial is multiplied by each term in the second polynomial. © 2001 McGraw-Hill Companies 404 MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS SECTION 6.4 405 (b) f(1) g(1) f(1) g(1) (1 3)(2(1) 5) 4(7) 28 (c) h(1) From part (a), we have h(x) 2x2 11x 15, so h(1) 2(1)2 11(1) 15 2 11 15 28 Again, we see that h(1) f(1) g(1). CHECK YOURSELF 4 Given f (x) 3x 2 and g(x) x 3, and letting h(x) f(x) g(x), find the following. (b) f(1) g(1) (a) h(x) (c) h(1) Certain products occur frequently enough in algebra that it is worth learning special formulas for dealing with them. Consider these products of two equal binomial factors. (a b)2 (a b)(a b) NOTE a 2ab b 2 2 and a2 2ab b2 a2 2ab b2 (a b)2 (a b)(a b) a2 2ab b2 are called perfect-square trinomials. We can summarize these statements as follows. Rules and Properties: Squaring a Binomial The square of a binomial has three terms, (1) the square of the first term, (2) twice the product of the two terms, and (3) the square of the last term. (a b)2 a2 2ab b2 and © 2001 McGraw-Hill Companies (a b)2 a2 2ab b2 Example 5 Squaring a Binomial Find each of the following binomial squares. NOTE Be sure to write out the expansion in detail. (1) (a) (x 5)2 x2 2(x)(5) 52 Square of first term Twice the product Square of last term of the two terms x2 10x 25 (2) 406 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS CAUTION (b) (2a 7)2 (2a)2 2(2a)(7) (7)2 4a2 28a 49 Be Careful! A very common mistake in squaring binomials is to forget the middle term! (y 7)2 is not equal to CHECK YOURSELF 5 y2 (7)2 Find each of the following binomial squares. The correct square is (a) (x 8)2 y2 14y 49 (b) (3x 5)2 The square of a binomial is always a trinomial. Another special product involves binomials that differ only in sign. It will be extremely important in your work later in this chapter on factoring. Consider the following: (a b)(a b) a2 ab ab b2 a2 b2 Rules and Properties: Product of Binomials Differing in Sign (a b)(a b) a2 b2 In words, the product of two binomials that differ only in the signs of their second terms is the difference of the squares of the two terms of the binomials. Example 6 Finding a Special Product Multiply. (a) (x 3)(x 3) x2 (3)2 x2 9 CAUTION The entire term 2x is squared, not just the x. (b) (2x 3y)(2x 3y) (2x)2 (3y)2 4x2 9y2 (c) (5a 4b2)(5a 4b2) (5a)2 (4b2)2 CHECK YOURSELF 6 Find each of the following products. (a) (y 5)(y 5) NOTE This format ensures that each term of one polynomial multiplies each term of the other. (b) (2x 3)(2x 3) (c) (4r 5s2)(4r 5s2) When multiplying two polynomials that don’t fit one of the special product patterns, there are two different ways to set up the multiplication. Example 7 will illustrate the vertical approach. © 2001 McGraw-Hill Companies 25a2 16b4 MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS SECTION 6.4 407 Example 7 Multiplying Polynomials Multiply 3x3 2x2 5 and 3x 2. Step 1 3x3 2x2 6x 2 3 Step 2 Step 3 5 3x 2 10 4x 5 3x 2 6x3 4x2 10 4 9x 6x3 15x Multiply by 2. 3x3 2x2 5 3x 2 6x3 4x2 10 4 9x 6x3 15x 9x4 4x2 15x 10 Multiply by 3x. Note that we align the terms in the partial product. 3x3 2x2 Add the partial products. CHECK YOURSELF 7 Find the following product, using the vertical method. (4x3 6x 7)(3x 2) A horizontal approach to the multiplication in Example 7 is also possible by the distributive property. As we see in Example 8, we first distribute 3x over the trinomial and then we distribute 2 over the trinomial. Example 8 Multiplying Polynomials Multiply (3x 2)(3x3 2x2 5), using a horizontal format. (3x 2)(3x3 2x2 5) Step 2 9x4 6x3 15x 6x3 4x2 10 NOTE Again, this ensures that each term of one polynomial multiplies each term of the other. © 2001 McGraw-Hill Companies Step 1 Step 1 Step 2 9x4 4x2 15x 10 Combine like terms. Write the product in descending form. CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS CHECK YOURSELF 8 Find the product of Check Yourself 7, using a horizontal format. Multiplication sometimes involves the product of more than two polynomials. In such cases, the associative property of multiplication allows us to regroup the factors to make the multiplication easier. Generally, we choose to start with the product of binomials. Example 9 illustrates this approach. Example 9 Multiplying Polynomials Find the products. (a) x(x 3)(x 3) x(x2 9) x3 9x (b) 2x(x 3)(2x 1) 2x(2x2 5x 3) 4x3 10x2 6x Find the product (x 3)(x 3). Then distribute x as the last step. Find the product of the binomials. Then distribute 2x. CHECK YOURSELF 9 Find each of the following products. (a) m(2m 3)(2m 3) (b) 3a(2a 5)(a 3) CHECK YOURSELF ANSWERS 1. 4. 5. 6. 8. (a) 36a6b3; (b) 35m4n6 2. h(x) 12x4 3x3 3. f(1) g(1) 15 h(1) 2 (a) h(x) 3x 7x 6; (b) f(1) g(1) 4; (c) h(1) 4 (a) x2 16x 64; (b) 9x2 30x 25 (a) y2 25; (b) 4x2 9; (c) 16r2 25s4 7. 12x4 8x3 18x2 9x 14 4 3 2 3 12x 8x 18x 9x 14 9. (a) 4m 9m; (b) 6a3 3a2 45a © 2001 McGraw-Hill Companies 408 Name 6.4 Exercises Section Date In exercises 1 to 6, find each product. 1. (4x)(5y) ANSWERS 2. (3m)(5n) 3. (6x2)(3x3) 4. (5y4)(3y2) 1. 2. 3. 4. 5. 5. (5r2s)(6r3s4) 6. (8a2b5)(3a3b2) 6. 7. In exercises 7 to 14, f(x) and g(x) are given. Let h(x) f(x) g(x). Find (a) h(x), (b) f(1) g(1), and (c) use the result of (a) to find h(1). 7. f(x) 3x and g(x) 2x2 3x 8. f(x) 4x and g(x) 2x2 7x 8. 9. 10. 9. f(x) 5x and g(x) 3x 5x 8 2 10. f(x) 2x and g(x) 7x 2x 2 2 11. 11. f(x) 4x3 and g(x) 9x2 3x 5 12. f(x) 2x3 and g(x) 2x3 4x 12. 13. 13. f(x) 3x3 and g(x) 5x2 4x 14. f(x) x2 and g(x) 7x3 5x2 14. 15. In exercises 15 to 24, find each product. 15. (x y)(x 3y) 16. 16. (x 3y)(x 5y) 17. 18. 17. (x 2y)(x 7y) 18. (x 7y)(x 3y) 19. © 2001 McGraw-Hill Companies 20. 19. (5x 7y)(5x 9y) 20. (3x 5y)(7x 2y) 21. 22. 21. (7x 5y)(7x 4y) 22. (9x 7y)(3x 2y) 23. (5x2 2y)(3x 2y2) 24. (6x2 5y2)(3x2 2y) 23. 24. 409 ANSWERS 25. In exercises 25 to 38, multiply polynomial expressions using the special product formulas. 26. 25. (x 5)2 26. (x 7)2 27. (2x 3)2 28. (5x 3)2 29. (4x 3y)2 30. (7x 5y)2 31. (4x 3y2)2 32. (3x3 7y)2 33. (x 3y)(x 3y) 34. (x 5y)(x 5y) 35. (2x 3y)(2x 3y) 36. (5x 3y)(5x 3y) 37. (4x2 3y)(4x2 3y) 38. (7x 6y2)(7x 6y2) 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. In exercises 39 to 42, multiply using the vertical format. 40. 39. (3x y)(x2 3xy y2) 40. (5x y)(x2 3xy y2) 41. (x 2y)(x2 2xy 4y2) 42. (x 3y)(x2 3xy 9y2) 41. 42. 43. In exercises 43 to 46, simplify each function. 44. 43. f(x) x(x 3)(x 1) 44. f(x) x(x 4)(x 2) 45. f(x) 2x(x 5)(x 4) 46. f(x) x2(x 4)(x2 5) 45. 46. Multiply the following. 48. 49. 47. 50. 2 3 3 5 x 2 2x 2 49. [x (y 2)][x (y 2)] 410 48. 3 4 4 5 x 3 3x 3 50. [x (3 y)][x (3 y)] © 2001 McGraw-Hill Companies 47. ANSWERS If the polynomial p(x) represents the selling price of an object, then the polynomial R(x), in which R(x) x p(x), is the revenue produced by selling x objects. Use this information to solve exercises 51 and 52. 51. If p(x) 100 0.2x, find R(x). 52. If p(x) 250 0.5x, find R(x). Find R(50). Find R(20). 52. 53. 54. 55. In exercises 53 to 56, label the statements as true or false. 53. (x y)2 x2 y2 51. 56. 54. (x y)2 x2 y2 57. 55. (x y) x 2xy y 2 2 2 56. (x y) x 2xy y 2 2 2 58. 57. Area. The length of a rectangle is given by 3x 5 centimeters (cm) and the width is 59. given by 2x 7 cm. Express the area of the rectangle in terms of x. 60. 58. Area. The base of a triangle measures 3y 7 inches (in.) and the height is 2y 3 in. 61. Express the area of the triangle in terms of y. 62. 2y 3 3y 7 59. Revenue. The price of an item is given by p 10 3x. If the revenue generated is found by multiplying the number of items x sold by the price of an item, find the polynomial that represents the revenue. 60. Revenue. The price of an item is given by p 100 2x2. Find the polynomial that represents the revenue generated from the sale of x items. 61. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are © 2001 McGraw-Hill Companies 5x 4 rows with 5x 4 trees in each row, how many trees are there in the orchard? 62. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the area of the square as a polynomial. 411 ANSWERS 63. 63. Area of a rectangle. The length and width of a rectangle are given by two consecutive odd integers. Write an expression for the area of the rectangle. 64. 64. Area of a rectangle. The length of a rectangle is 6 less than three times the width. 65. Write an expression for the area of the rectangle. 65. Work with another student to complete this table and write the polynomial. A paper 66. box is to be made from a piece of cardboard 20 inches (in.) wide and 30 in. long. The box will be formed by cutting squares out of each of the four corners and folding up the sides to make a box. 67. 30 in. x 20 in. If x is the dimension of the side of the square cut out of the corner, when the sides are folded up, the box will be x inches tall. You should use a piece of paper to try this to see how the box will be made. Complete the following chart. Length of Side of Corner Square Length of Box Width of Box Depth of Box Volume of Box 1 in. 2 in. 3 in. n in. 66. Complete the following statement: (a b)2 is not equal to a2 b2 because. . . . But, wait! Isn’t (a b)2 sometimes equal to a2 b2? What do you think? 67. Is (a b)3 ever equal to a3 b3? Explain. 412 © 2001 McGraw-Hill Companies Write general formulas for the width, length, and height of the box and a general formula for the volume of the box, and simplify it by multiplying. The variable will be the height, the side of the square cut out of the corners. What is the highest power of the variable in the polynomial you have written for the volume? Extend the table to decide what the dimensions are for a box with maximum volume. Draw a sketch of this box and write in the dimensions. ANSWERS 68. In the following figures, identify the length and the width of the square, and then find the area. a 68. 69. b Length ________ 70. a Width ________ b Area ________ 2 x 71. 72. 73. Length ________ x Width ________ 2 Area ________ 69. The square shown is x units on a side. The area is _______. x x Draw a picture of what happens when the sides are doubled. The area is _______. Continue the picture to show what happens when the sides are tripled. The area is _______. If the sides are quadrupled, the area is _______. In general, if the sides are multiplied by n, the area is _______. If each side is increased by 3, the area is increased by _______. If each side is decreased by 2, the area is decreased by _______. © 2001 McGraw-Hill Companies In general, if each side is increased by n, the area is increased by _______, and if each side is decreased by n, the area is decreased by _______. For each of the following problems, let x represent the number, then write an expression for the product. 70. The product of 6 more than a number and 6 less than that number. 71. The square of 5 more than a number. 72. The square of 4 less than a number. 73. The product of 5 less than a number and 5 more than that number. 413 ANSWERS 74. Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of the following products. 75. 76. 74. (49)(51) 75. (27)(33) 76. (34)(26) 77. (98)(102) 78. (55)(65) 79. (64)(56) 77. 78. 79. Answers 1. 20xy 3. 18x5 5. 30r5s5 7. (a) 6x3 9x2; (b) 3; (c) 3 3 2 9. (a) 15x 25x 40x; (b) 0; (c) 0 11. (a) 36x5 12x4 20x3; (b) 28; (c) 28 5 4 2 13. (a) 15x 12x ; (b) 3; (c) 3 15. x 4xy 3y2 2 2 2 17. x 5xy 14y 19. 25x 80xy 63y2 21. 49x2 63xy 20y2 3 2 2 3 2 23. 15x 10x y 6xy 4y 25. x 10x 25 27. 4x2 12x 9 2 2 2 2 4 29. 16x 24xy 9y 31. 16x 24xy 9y 33. x2 9y2 2 2 4 2 3 2 35. 4x 9y 37. 16x 9y 39. 3x 8x y 6xy2 y3 3 3 3 2 41. x 8y 43. x 2x 3x 45. 2x3 2x2 40x x2 11x 4 49. x2 y2 4y 4 51. 100x 0.2x2, 4500 3 45 15 53. False 55. True 57. 6x2 11x 35 cm2 59. 10x 3x2 2 2 61. 25x 40x 16 63. x(x 2) or x 2x 65. 47. 67. 73. x2 25 75. 891 77. 9996 79. 3584 © 2001 McGraw-Hill Companies 71. x2 10x 25 69. x2; 4x2; 9x2; 16x2; n2x2; 6x 9; 4x 4; 2xn n2; 2xn n2 414