5.4 E x a m p l e 1 362SECTION 5.4 OBJECTIVES 1. Find the product of a monomial and a polynomial 2. Find the product of two polynomials 3. Square a polynomial 4. Find the product of two binomials that differ only in sign Once again we have used the commutative and associative properties to rewrite the problem. Multiplying of Polynomials and Special Products You have already had some experience in multiplying polynomials. In Section 5.1 we stated the product rule for exponents and used that rule to find the product of two monomials. Let s review briefly. Multiplying Monomials Multiply 3x 2 y and 2x 3 y 5. Write To Find the Product of Monomials Step 1 Step 2 Multiply the coefficients. Use the product rule for exponents to combine the variables: CHECK YOURSELF 1 (3x 2 y)(2x 3 y 5 ) Multiply the coefficients. 6x 5 y 6 (a) (5a 2 b)(3a 2 b 4 ) (b) ( 3xy)(4x 3 y 5 ) ax m bx n abx m n Let s look at an example in which we multiply two monomials. (3 2)(x 2 x 3 )(y y 5 ) Add the exponents.
Section 5.4 Multiplying of Polynomials and Special Products 363 You might want to review Section 0.5 before going on. Our next task is to find the product of a monomial and a polynomial. Here we use the distributive property, which we introduced in Section 0.5. That property leads us to the following rule for multiplication. To Multiply a Polynomial by a Monomial Distributive property: a(b c) ab ac Use the distributive property to multiply each term of the polynomial by the monomial and simplify the result. E x a m p l e 2 Multiplying a Monomial and a Binomial Note: With practice you will do this step mentally. (a) Multiply 2x 3 by x. Write x(2x 3) x 2x x 3 2x 2 3x (b) Multiply 2a 3 4a by 3a 2. Write Multiply x by 2x and then by 3, the terms of the polynomial. That is, distribute the multiplication over the sum. CHECK YOURSELF 2 (a) 2y(y 2 3y) (b) 3w 2 (2w 3 5w) 3a 2 (2a 3 4a) 3a 2 2a 3 3a 2 4a 6a 5 12a 3 The patterns of Example 2 extend to any number of terms. E x a m p l e 3 Multiplying a Monomial and a Polynomial Multiply the following. (a) 3x(4x 3 5x 2 2) 3x 4x 3 3x 5x 2 3x 2 12x 4 15x 3 6x
364 Chapter 5 Polynomials Again we have shown all the steps of the process. With practice you can write the product directly, and you should try to do so. (b) 5c(4c 2 8c) ( 5c)(4c 2 ) ( 5c)(8c) 20c 3 40c 2 (c) 3c 2 d 2 (7cd 2 5c 2 d 3 ) 3c 2 d 2 7cd 2 (3c 2 d 2 )(5c 2 d 3 ) 21c 3 d 4 15c 4 d 5 CHECK YOURSELF 3 (a) 3(5a 2 2a 7) (b) 4x 2 (8x 3 6) (c) 5m(8m 2 5m) (d) 9a 2 b(3a 3 b 6a 2 b 4 ) E x a m p l e 4 Multiplying Binomials (a) Multiply x 2 by x 3. We can think of x 2 as a single quantity and apply the distributive property. Note that this ensures that each term, x and 2, of the first binomial is multiplied by each term, x and 3, of the second binomial. (x 2)(x 3) (x 2)x (x 2)3 x x 2 x x 3 2 3 x 2 2x 3x 6 x 2 5x 6 Multiply x 2 by x and then by 3. (b) Multiply a 3 by a 4. (a 3)(a 4) (Think of a 3 as a single quantity and distribute.) (a 3)a (a 3)(4) a a 3 a [(a 4) (3 4)] a 2 3a (4a 12) Note that the parentheses are needed here because a minus sign precedes the binomial. CHECK YOURSELF 4 (a) (x 4)(x 5) (b) (y 5)(y 6)
Section 5.4 Multiplying of Polynomials and Special Products 365 Fortunately, there is a pattern to this kind of multiplication that allows you to write the product of the two binomials directly without going through all these steps. We call it the FOIL method of multiplying. The reason for this name will be clear as we look at the process in more detail. To multiply (x 2)(x 3): Remember this by F! Remember this by O! Remember this by I! Remember this by L! 1. (x 2)(x 3) x x Find the product of the first terms of the factors. 2. (x 2)(x 3) x 3 Find the product of the outer terms. 3. (x 2)(x 3) 2 x Find the product of the inner terms. 4. (x 2)(x 3) 2 3 Find the product of the last terms. Combining the four steps, we have It s called FOIL to give you an easy way of remembering the steps: First, Outer, Inner, and Last. (x 2)(x 3) x 2 3x 2x 6 x 2 5x 6 With practice, the FOIL method will let you write the products quickly and easily. Consider Example 5, which illustrates this approach. E x a m p l e 5 Using the FOIL Method Find the following products, using the FOIL method. F L x x 4 5 (a) (x 4)(x 5) 4x I When possible, you should combine the outer and inner products mentally and write just the final product. 5x O x 2 5x 4x 20 F O I L x 2 9x 20
366 Chapter 5 Polynomials F L x x ( 7)(3) b) (x 7)(x 3) 7x I Combine the outer and inner products as 4x. 3x O x 2 4x 21 CHECK YOURSELF 5 (a) (x 6)(x 7) (b) (x 3)(x 5) (c) (x 2)(x 8) Using the FOIL method, you can also find the product of binomials with leading coefficients other than 1 or with more than one variable. E x a m p l e 6 Using the FOIL Method Find the following products, using the FOIL method. 12x 2 6 (a) (4x 3)(3x 2) 9x Combine: 9x 9x x 8x 12x 2 x 6 6x 2 35y 2 (b) (3x 5y)(2x 7y) 10xy 21xy 6x 2 31xy 35y 2 Combine: 10xy 21xy 31xy
Section 5.4 Multiplying of Polynomials and Special Products 367 The following rule summarizes our work in multiplying binomials. To Multiply Two Binomials Step 1 Step 2 Step 3 Find the first term of the product of the binomials by multiplying the first terms of the binomials (F). Find the middle term of the product as the sum of the outer and inner products (O I). Find the last term of the product by multiplying the last terms of the binomials (L). CHECK YOURSELF 6 (a) (5x 2)(3x 7) (b) (4a 3b)(5a 4b) (c) (3m 5n)(2m 3n) The FOIL method works well when multiplying any two binomials. But what if one of the factors has three or more terms? The vertical format, shown in Example 7, works for factors with any number of terms. E x a m p l e 7 Using the Vertical Method Multiply x 2 5x 8 by x 3. Step 1 x 2 5x 8 x 3 Multiply each term of x 2 5x 8 by 3. 3x 2 15x 24 Step 2 x 2 5x 8 x 3 Now multiply each term by x. 3x 2 15x 24 x 3 5x 2 8x Note that this line is shifted over so that like terms are in the same columns. Note: Using this vertical method ensures that each term of one factor multiples each term of the other. That s why it works! Step 3 x 2 5x 8 x 3 3x 2 15x 24 x 3 5x 2 8x Now add to combine like terms to write the x 3 2x 2 7x 24 product.
368 Chapter 5 Polynomials CHECK YOURSELF 7 Multiply 2x 2 5x 3 by 3x 4. Certain products occur frequently enough in algebra that it is worth learning special formulas for dealing with them. First, let s look at the square of a binomial, which is the product of two equal binomial factors. (x y) 2 (x y)(x y) x 2 2xy y 2 (x y) 2 (x y)(x y) x 2 2xy y 2 The patterns above lead us to the following rule. To Square a Binomial Step 1 Step 2 Step 3 Find the first term of the square by squaring the first term of the binomial. Find the middle term of the square as twice the product of the two terms of the binomial. Find the last term of the square by squaring the last term of the binomial. E x a m p l e 8 Squaring a Binomial (a) (x 3) 2 x 2 2 x 3 3 2 Caution! A very common mistake in squaring binomials is to forget the middle term. Twice the Square of product of Square of first term the two terms the last term x 2 6x 9 (b) (3a 4b) 2 (3a) 2 2(3a)(4b) (4b) 2 9a 2 24ab 16b 2 (c) (y 5) 2 y 2 2 y ( 5) ( 5) 2 y 2 10y 25
Section 5.4 Multiplying of Polynomials and Special Products 369 Again we have shown all the steps. With practice you can write just the square. (d) (5c 3d) 2 (5c) 2 2(5c)( 3d) ( 3d) 2 25c 2 30cd 9d 2 CHECK YOURSELF 8 (a) (2x 1) 2 (b) (4x 3y) 2 E x a m p l e 9 Squaring a Binomial Find (y 4) 2. The correct square is (y 4) 2 is not equal to y 2 4 2 or y 2 16 You should see that (2 3) 2 2 2 3 2 because 5 2 4 9 CHECK YOURSELF 9 (y 4) 2 y 2 8y 16 The middle term is twice the product of y and 4. (a) (x 5) 2 (b) (3a 2) 2 (c) (y 7) 2 (d) (5x 2y) 2 A second special product will be very important in the next chapter, which deals with factoring. Suppose the form of a product is Let s see what happens when we multiply. (x y)(x y) The two factors differ only in sign. (x y)(x y) x 2 xy xy y 2 x 2 y 2 0
370 Chapter 5 Polynomials Since the middle term becomes 0, we have the following rule. Special Product The product of two binomials that differ only in the sign between the terms is the square of the first term minus the square of the second term. Let s look at the application of this rule in Example 10. E x a m p l e 1 0 Multiplying Polynomials Multiply each pair of factors. (a) (x 5)(x 5) x 2 5 2 Square of the first term x 2 25 Square of the second term Note: (2y) 2 (2y)(2y) 4y 2 (b) (x 2y)(x 2y) x 2 (2y) 2 Square of Square of the first term the second term x 2 4y 2 (c) (3m n)(3m n) 9m 2 n 2 (d) (4a 3b)(4a 3b) 16a 2 9b 2 CHECK YOURSELF 10 Find the products. (a) (a 6)(a 6) (b) (x 3y)(x 3y) (c) (5n 2p)(5n 2p) (d) (7b 3c)(7b 3c) When finding the product of three or more factors, it is useful to first look for the pattern in which two binomials differ only in their sign. Finding this product first will make it easier to find the product of all the factors.
Section 5.4 Multiplying of Polynomials and Special Products 371 E x a m p l e 1 1 Multiplying Polynomials (a) x(x 3)(x 3) x(x 2 9) x 3 9x (b) (x 1)(x 5)(x 5) (x 1)(x 2 25) x 3 x 2 25x 25 These binomials differ only in the sign. These binomials differ only in the sign. With two binomials, use the FOIL method. (c) (2x 1)(x 3)(2x 1) (2x 1) (x 3) (2x 1) These two binomials differ only in the sign of the second term. We can use the commutative (x 3)(2x 1)(2x 1) property to rearrange the terms. (x 3)(4x 2 1) 4x 3 12x 2 x 3 CHECK YOURSELF 11 (a) 3x(x 5)(x 5) (b) (x 4)(2x 3)(2x 3) (c) (x 7)(3x 1)(x 7) CHECK YOURSELF ANSWERS 1. (a) 15a 4 b 5 ; (b) 12x 4 y 6. 2. (a) 2y 3 6y 2 ; (b) 6w 5 15w 3. 3. (a) 15a 2 6a 21; (b) 32x 5 24x 2 ; (c) 40m 3 25m 2 ; (d) 27a 5 b 2 54a 4 b 5. 4. (a) x 2 9x 20; (b) y 2 y 30. 5. (a) x 2 13x 42; (b) x 2 2x 15; (c) x 2 10x 16. 6. (a) 15x 2 29x 14; (b) 20a 2 31ab 12b 2 ; (c) 6m 2 19mn 15n 2. 7. 6x 3 7x 2 11x 12. 8. (a) 4x 2 4x 1; (b) 16x 2 24xy 9y 2. 9. (a) x 2 10x 25; (b) 9a 2 12a 4; (c) y 2 14y 49; (d) 25x 2 20xy 4y 2. 10. (a) a 2 36; (b) x 2 9y 2 ; (c) 25n 2 4p 2 ; (d) 49b 2 9c 2. 11. (a) 3x 3 75x; (b) 4x 3 16x 2 9x 36; (c) 3x 3 x 2 147x 49.
E x e r c i s e s 5.4 1. 15x 5 2. 28a 11 3. 28b 10 4. 56y 10 5. 40p 13 6. 54m 15 7. 12m 6 8. 15r 8 9. 32x 5 y 3 10. 21r 6 s 7 11. 6m 9 n 3 12. 42a 7 b 6 13. 10x 30 14. 28b 20 15. 12a 2 15a 16. 10x 2 35x 17. 12s 4 21s 3 18. 27a 5 45a 3 19. 8x 3 4x 2 2x 20. 20m 4 15m 3 10m 21. 6x 3 y 2 3x 2 y 3 15x 2 y 2 22. 5a 2 b 3 15a 2 b 2 25ab 3 23. 18m 4 n 2 12m 3 n 2 6m 3 n 3 24. 16p 2 q 3 24p 2 q 2 40pq 3 25. x 2 5x 6 26. a 2 10a 21 27. m 2 14m 45 28. b 2 12b 35 29. p 2 p 56 30. x 2 x 90 31. w 2 30w 200 32. s 2 20s 96 33. 3x 2 29x 40 34. 4w 2 13w 35 35. 6x 2 x 12 36. 15a 2 38a 7 37. 12a 2 31ab 9b 2 38. 21s 2 47st 24t 2 39. 21p 2 13pq 20q 2 1. (5x 2 )(3x 3 ) 2. (7a 5 )(4a 6 ) 3. ( 2b 2 )(14b 8 ) 4. (14y 4 )( 4y 6 ) 5. ( 10p 6 )( 4p 7 ) 6. ( 6m 8 )(9m 7 ) 7. (4m 5 )( 3m) 8. ( 5r 7 )( 3r) 9. (4x 3 y 2 )(8x 2 y) 10. ( 3r 4 s 2 )( 7r 2 s 5 ) 11. ( 3m 5 n 2 )(2m 4 n) 12. (7a 3 b 5 )( 6a 4 b) 13. 5(2x 6) 14. 4(7b 5) 15. 3a(4a 5) 16. 5x(2x 7) 17. 3s 2 (4s 2 7s) 18. 9a 2 (3a 3 5a) 19. 2x(4x 2 2x 1) 20. 5m(4m 3 3m 2 2) 21. 3xy(2x 2 y xy 2 5xy) 22. 5ab 2 (ab 3a 5b) 23. 6m 2 n(3m 2 n 2mn mn 2 ) 24. 8pq 2 (2pq 3p 5q) 25. (x 3)(x 2) 26. (a 3)(a 7) 27. (m 5)(m 9) 28. (b 7)(b 5) 29. (p 8)(p 7) 30. (x 10)(x 9) 31. (w 10)(w 20) 32. (s 12)(s 8) 33. (3x 5)(x 8) 34. (w 5)(4w 7) 35. (2x 3)(3x 4) 36. (5a 1)(3a 7) 37. (3a b)(4a 9b) 38. (7s 3t)(3s 8t) 39. (3p 4q)(7p 5q) 40. (5x 4y)(2x y) 41. (2x 5y)(3x 4y) 42. (4x 5y)(4x 3y) 40. 10x 2 13xy 4y 2 41. 6x 2 23xy 20y 2 42. 16x 2 8xy 15y 2 372
Section 5.4 Multiplying of Polynomials and Special Products 373 43. x 2 10x 25 44. y 2 18y 81 45. w 2 12w 36 46. a 2 16a 64 47. z 2 24z 144 48. p 2 40p 400 49. 4a 2 4a 1 50. 9x 2 12x 4 51. 36m 2 12m 1 52. 49b 2 28b 4 53. 9x 2 6xy y 2 54. 25m 2 10mn n 2 55. 4r 2 20rs 25s 2 56. 9a 2 24ab 16b 2 Find each of the following squares. 43. (x 5) 2 44. (y 9) 2 45. (w 6) 2 46. (a 8) 2 47. (z 12) 2 48. (p 20) 2 49. (2a 1) 2 50. (3x 2) 2 51. (6m 1) 2 52. (7b 2) 2 53. (3x y) 2 54. (5m n) 2 55. (2r 5s) 2 56. (3a 4b) 2 57. (8a 9b) 2 58. (7p 6q) 2 59. x 1 2 2 60. w 1 4 2 57. 64a 2 144ab 81b 2 58. 49p 2 84pq 36q 2 59. x 2 x 1 4 60. w 2 1 2 w 1 16 61. x 2 36 62. y 2 64 63. m 2 144 64. w 2 100 65. x 2 1 4 66. x2 4 9 67. p 2 0.16 68. m 2 0.36 69. a 2 9b 2 70. p 2 16q 2 Find each of the following products. 61. (x 6)(x 6) 62. (y 8)(y 8) 63. (m 12)(m 12) 64. (w 10)(w 10) 65. x 1 2 x 1 2 66. x 2 3 x 2 3 67. (p 0.4)(p 0.4) 68. (m 0.6)(m 0.6) 69. (a 3b)(a 3b) 70. (p 4q)(p 4q) 71. (4r s)(4r s) 72. (7x y)(7x y) 73. (8w 5z)(8w 5z) 74. (7c 2d)(7c 2d) 75. (5x 9y)(5x 9y) 76. (6s 5t)(6s 5t) 71. 16r 2 s 2 72. 49x 2 y 2 73. 64w 2 25z 2 74. 49c 2 4d 2 75. 25x 2 81y 2 76. 36s 2 25t 2
374 Chapter 5 Polynomials 77. 24x 3 10x 2 4x 78. 12x 3 3x 79. 80a 3 45a 80. 54m 3 162m 2 84m 81. 60s 3 39s 2 6s 82. 28w 3 63w 83. x 3 4x 2 x 6 84. y 3 3y 2 10y 24 85. a 3 3a 2 3a 1 86. x 3 3x 2 3x 1 2 87. x 1 1x 4 3 45 1 5 2 88. x 2 9 9 x 4 80 2 0 89. x 2 y 2 4y 4 90. x 2 y 2 6y 9 91. False 92. False 93. True 94. True 95. 6x 2 11x 35 cm 2 96. 3y 2 5 2 y 2 1 in 2 2 97. 10x 3x 2 98. 100x 2x 3 99. 25x 2 40x 16 77. 2x(3x 2)(4x 1) 78. 3x(2x 1)(2x 1) 79. 5a(4a 3)(4a 3) 80. 6m(3m 2)(3m 7) 81. 3s(5s 2)(4s 1) 82. 7w(2w 3)(2w 3) 83. (x 2)(x 1)(x 3) 84. (y 3)(y 2)(y 4) 85. (a 1) 3 86. (x 1) 3 Multiply the following. 87. x 2 2 3 2 x 2 3 5 88. x 3 3 4 3 x 3 4 5 89. [x (y 2)][x (y 2)] 90. [x (3 y)][x (3 y)] Label the following as true or false. 91. (x y) 2 x 2 y 2 92. (x y) 2 x 2 y 2 93. (x y) 2 x 2 2xy y 2 94. (x y) 2 x 2 2xy y 2 95. Length. The length of a rectangle is given by 3x 5 centimeters (cm) and the width is given by 2x 7 cm. Express the area of the rectangle in terms of x. 96. Area. The base of a triangle measures 3y 7 inches (in.) and the height is 2y 3 in. Express the area of the triangle in terms of y. 97. 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 which represents the revenue. 98. Revenue. The price of an item is given by p 100 2x 2. Find the polynomial that represents the revenue generated from the sale of x items. 99. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are 5x 4 rows with 5x 4 trees in each row, how many trees are there in the orchard?
Section 5.4 Multiplying of Polynomials and Special Products 375 100. 9x 2 12x 4 cm 2 101. x(x 2) or x 2 2x 102. x(3x 6) or 3x 2 6x 100. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the area of the square as a polynomial. 101. 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. 102. Area of a rectangle. The length of a rectangle is 6 less than three times the width. Write an expression for the area of the rectangle. 103. Work with another student to complete this table and write the polynomial. A paper 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. x 30 in. 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 Length of Width of Depth of Volume of Corner Square Box Box Box Box 1 in. 2 in. 3 in. n in. 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. 104. Complete the following statement: (a b) 2 is not equal to a 2 b 2 because.... But, wait! Isn t (a b) 2 sometimes equal to a 2 b 2? What do you think? 105. Is (a b) 3 ever equal to a 3 b 3? Explain.
376 Chapter 5 Polynomials 106. In the following figures, identify the length and the width of the square: a b Length a b Width Area x 2 Length x Width 2 Area x x 107. The square shown is x units on a side. The area is. 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. 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. 108. x 2 36 109. x 2 10x 25 110. x 2 8x 16 111. x 2 25 112. 2499 113. 891 114. 884 115. 9996 116. 3575 117. 3584 For each of the following problems, let x represent the number, then write an expression for the product. 108. The product of 6 more than a number and 6 less than that number 109. The square of 5 more than a number 110. The square of 4 less than a number 111. The product of 5 less than a number and 5 more than that number Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of the following products. 112. (49)(51) 113. (27)(33) 114. (34)(26) 115. (98)(102) 116. (55)(65) 117. (64)(56)