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SECT ION 5.4 5.4 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 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. To Find the Product of Monomials Step 1 Multiply the coefficients. Step 2 Use the product rule for exponents to combine the variables: axm bxn abxmn Let’s look at an example in which we multiply two monomials. Example 1 Multiplying Monomials Multiply 3x2y and 2x3y5. Write Once again we have used the commutative and associative properties to rewrite the problem. (3x2y)(2x3y5) (3 2)(x2 x3)(y y5) 6x y 5 6 ✓ CHECK YOURSELF 1 ■ Multiply. (a) (5a2b)(3a2b4) 362 (b) (3xy)(4x3y5) Multiply the coefficients. Add the exponents. Section 5.4 You might want to review Section 0.5 before going on. ■ Multiplying of Polynomials and Special Products 363 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 Example 2 Use the distributive property to multiply each term of the polynomial by the monomial and simplify the result. Multiplying a Monomial and a Binomial (a) Multiply 2x 3 by x. Write x(2x 3) Note: With practice you will do this step mentally. x 2x x 3 Multiply x by 2x and then by 3, the terms of the polynomial. That is, “distribute” the multiplication over the sum. 2x2 3x (b) Multiply 2a3 4a by 3a2. Write 3a2(2a3 4a) 3a2 2a3 3a2 4a 6a5 12a3 ✓ CHECK YOURSELF 2 ■ Multiply. (a) 2y(y2 3y) (b) 3w2(2w3 5w) The patterns of Example 2 extend to any number of terms. Example 3 Multiplying a Monomial and a Polynomial Multiply the following. (a) 3x(4x3 5x2 2) 3x 4x3 3x 5x2 3x 2 12x4 15x3 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(4c2 8c) (5c)(4c2) (5c)(8c) 20c3 40c2 (c) 3c2d2(7cd2 5c2d3) 3c2d2 7cd2 (3c2d2)(5c2d3) 21c3d4 15c4d5 ✓ CHECK YOURSELF 3 ■ Multiply. Example 4 (a) 3(5a2 2a 7) (b) 4x2(8x3 6) (c) 5m(8m2 5m) (d) 9a2b(3a3b 6a2b4) Multiplying Binomials (a) Multiply x 2 by x 3. We can think of x 2 as a single quantity and apply the distributive property. (x 2)(x 3) 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. Multiply x 2 by x and then by 3. (x 2)x (x 2)3 xx2xx323 x2 2x 3x 6 x2 5x 6 (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)] a2 3a (4a 12) Note that the parentheses are needed here because a minus sign precedes the binomial. ✓ CHECK YOURSELF 4 ■ Multiply. (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! 1. (x 2)(x 3) xx Remember this by O! 2. Find the product of the first terms of the factors. (x 2)(x 3) x3 Remember this by I! 3. Find the product of the outer terms. (x 2)(x 3) 2x Remember this by L! 4. Find the product of the inner terms. (x 2)(x 3) 23 Find the product of the last terms. Combining the four steps, we have (x 2)(x 3) It’s called FOIL to give you an easy way of remembering the steps: First, Outer, Inner, and Last. Example 5 x2 3x 2x 6 x2 5x 6 With practice, the FOIL method will let you write the products quickly and easily. Consider Example 5, which illustrates this approach. Using the FOIL Method Find the following products, using the FOIL method. F xx L 45 (a) (x 4)(x 5) 4x I 5x O When possible, you should combine the outer and inner products mentally and write just the final product. x 5x 4x 20 2 F O I x 9x 20 2 L 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 4x 21 2 ✓ CHECK YOURSELF 5 ■ Multiply. (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. Example 6 Using the FOIL Method Find the following products, using the FOIL method. 6 12x2 (a) (4x 3)(3x 2) 9x Combine: 9x 9x x 8x 12x2 x 6 6x2 35y2 (b) (3x 5y)(2x 7y) 10xy 21xy Combine: 10xy 21xy 31xy 6x2 31xy 35y2 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 Find the first term of the product of the binomials by multiplying the first terms of the binomials (F). Step 2 Find the middle term of the product as the sum of the outer and inner products (O I). Step 3 Find the last term of the product by multiplying the last terms of the binomials (L). ✓ CHECK YOURSELF 6 ■ Multiply. (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. Example 7 Using the Vertical Method Multiply x2 5x 8 by x 3. Step 1 Step 2 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 x2 5x 8 x 3 ——————— 2 3x 15x 24 x2 5x 8 x 3 ——————— 2 3x 15x 24 x3 5x2 8x —————————— x2 5x 8 x 3 ——————— 2 3x 15x 24 x3 5x2 8x —————————— x3 2x2 7x 24 Multiply each term of x2 5x 8 by 3. Now multiply each term by x. Note that this line is shifted over so that like terms are in the same columns. Now add to combine like terms to write the product. 368 Chapter 5 ■ Polynomials ✓ CHECK YOURSELF 7 ■ Multiply 2x2 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) x2 2xy y2 (x y)2 (x y)(x y) x2 2xy y2 The patterns above lead us to the following rule. To Square a Binomial Step 1 Find the first term of the square by squaring the first term of the binomial. Step 2 Find the middle term of the square as twice the product of the two terms of the binomial. Step 3 Find the last term of the square by squaring the last term of the binomial. Squaring a Binomial Example 8 (a) (x 3)2 x2 2 x 3 32 Caution ! A very common mistake in squaring binomials is to forget the middle term. Square of first term Twice the product of the two terms Square of the last term x2 6x 9 (b) (3a 4b)2 (3a)2 2(3a)(4b) (4b)2 9a2 24ab 16b2 (c) (y 5)2 y2 2 y (5) (5)2 y2 10y 25 Section 5.4 Again we have shown all the steps. With practice you can write just the square. ■ Multiplying of Polynomials and Special Products 369 (d) (5c 3d)2 (5c)2 2(5c)(3d) (3d)2 25c2 30cd 9d2 ✓ CHECK YOURSELF 8 ■ Multiply. (a) (2x 1)2 Example 9 (b) (4x 3y)2 Squaring a Binomial Find (y 4)2. (y 4)2 is not equal to y2 42 or y2 16 The correct square is (y 4)2 y2 8y 16 You should see that (2 3)2 22 32 because 52 4 9 The middle term is twice the product of y and 4. ✓ CHECK YOURSELF 9 ■ Multiply. (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 (x y)(x y) The two factors differ only in sign. Let’s see what happens when we multiply. (x y)(x y) x2 xy xy y2 0 x y 2 2 370 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. Example 10 Multiplying Polynomials Multiply each pair of factors. (a) (x 5)(x 5) x2 52 Square of the first term Square of the second term x2 25 Note: (2y)2 (2y)(2y) 4y2 (b) (x 2y)(x 2y) x2 (2y)2 Square of the first term Square of the second term x2 4y2 (c) (3m n)(3m n) 9m2 n2 (d) (4a 3b)(4a 3b) 16a2 9b2 ✓ 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 Example 11 Multiplying of Polynomials and Special Products ■ 371 Multiplying Polynomials (a) x(x 3)(x 3) TThese binomials differ only in the sign. x(x 9) 2 x3 9x (b) (x 1)(x 5)(x 5) These binomials differ only in the sign. (x 1)(x2 25) With two binomials, use the FOIL method. x x 25x 25 3 2 (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 property to rearrange the terms. (x 3)(2x 1)(2x 1) (x 3)(4x2 1) 4x3 12x2 x 3 ✓ CHECK YOURSELF 11 ■ Multiply. (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) 15a4b5; (b) 12x4y6. 2. (a) 2y3 6y2; (b) 6w5 15w3. 3. (a) 15a2 6a 21; (b) 32x5 24x2; (c) 40m3 25m2; (d) 27a5b2 54a4b5. 4. (a) x2 9x 20; (b) y2 y 30. (c) x2 10x 16. 5. (a) x2 13x 42; (b) x2 2x 15; 6. (a) 15x2 29x 14; (b) 20a2 31ab 12b2; (c) 6m2 19mn 15n2. (b) 16x2 24xy 9y2. 7. 6x3 7x2 11x 12. 9. (a) x2 10x 25; (b) 9a2 12a 4; (c) y2 14y 49; (d) 25x2 20xy 4y2. (c) 25n2 4p2; (d) 49b2 9c2. (c) 3x3 x2 147x 49. 8. (a) 4x2 4x 1; 10. (a) a2 36; (b) x2 9y2; 11. (a) 3x3 75x; (b) 4x3 16x2 9x 36; Exercises 1. 15x5 2. 28a11 3. 28b10 4. 56y10 13 ■ 5.4 Multiply. 1. (5x2)(3x3) 2. (7a5)(4a6) 3. (2b2)(14b8) 4. (14y4)(4y6) 5. (10p6)(4p7) 6. (6m8)(9m7) 7. (4m5)(3m) 8. (5r7)(3r) 15 6. 54m 5. 40p 6 8 7. 12m 8. 15r 5 3 10. 21r6s7 9. 32x y 11. 6m9n3 12. 42a7b6 13. 10x 30 14. 28b 20 15. 12a2 15a 16. 10x2 35x 11. (3m5n2)(2m4n) 12. (7a3b5)(6a4b) 17. 12s4 21s3 18. 27a5 45a3 13. 5(2x 6) 14. 4(7b 5) 15. 3a(4a 5) 16. 5x(2x 7) 17. 3s2(4s2 7s) 18. 9a2(3a3 5a) 19. 2x(4x2 2x 1) 20. 5m(4m3 3m2 2) 21. 3xy(2x2y xy2 5xy) 22. 5ab2(ab 3a 5b) 23. 6m2n(3m2n 2mn mn2) 24. 8pq2(2pq 3p 5q) 19. 8x3 4x2 2x 4 3 20. 20m 15m 10m 21. 6x3y2 3x2y3 15x2y2 2 3 2 2 3 22. 5a b 15a b 25ab 4 2 3 2 3 3 23. 18m n 12m n 6m n 24. 16p2q3 24p2q2 40pq3 25. x2 5x 6 26. a2 10a 21 27. m2 14m 45 28. b2 12b 35 9. (4x3y2)(8x2y) 10. (3r4s2)(7r2s5) Multiply. 25. (x 3)(x 2) 26. (a 3)(a 7) 31. w2 30w 200 27. (m 5)(m 9) 28. (b 7)(b 5) 32. s2 20s 96 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) 29. p2 p 56 30. x2 x 90 2 33. 3x 29x 40 2 34. 4w 13w 35 35. 6x2 x 12 36. 15a2 38a 7 37. 12a2 31ab 9b2 38. 21s2 47st 24t2 39. 21p2 13pq 20q2 40. 10x2 13xy 4y2 41. 6x2 23xy 20y2 42. 16x2 8xy 15y2 372 Section 5.4 ■ Multiplying of Polynomials and Special Products 43. x2 10x 25 Find each of the following squares. 44. y2 18y 81 43. (x 5)2 44. (y 9)2 45. (w 6)2 46. (a 8)2 47. z2 24z 144 47. (z 12)2 48. (p 20)2 48. p2 40p 400 49. (2a 1)2 50. (3x 2)2 51. (6m 1)2 52. (7b 2)2 51. 36m2 12m 1 53. (3x y)2 54. (5m n)2 52. 49b2 28b 4 55. (2r 5s)2 56. (3a 4b)2 57. (8a 9b)2 58. (7p 6q)2 45. w2 12w 36 46. a2 16a 64 49. 4a2 4a 1 50. 9x2 12x 4 53. 9x2 6xy y2 54. 25m2 10mn n2 55. 4r2 20rs 25s2 56. 9a2 24ab 16b2 1 59. x 2 2 1 60. w 4 2 57. 64a2 144ab 81b2 58. 49p2 84pq 36q2 Find each of the following products. 1 59. x2 x 4 61. (x 6)(x 6) 62. (y 8)(y 8) 1 1 60. w2 w 2 16 63. (m 12)(m 12) 64. (w 10)(w 10) 61. x2 36 1 1 65. x x 2 2 2 2 66. x x 3 3 64. w2 100 1 4 65. x2 66. x2 4 9 67. (p 0.4)(p 0.4) 68. (m 0.6)(m 0.6) 69. (a 3b)(a 3b) 70. (p 4q)(p 4q) 67. p2 0.16 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) 62. y2 64 63. m2 144 2 68. m 0.36 69. a2 9b2 70. p2 16q2 71. 16r2 s2 72. 49x2 y2 73. 64w2 25z2 74. 49c2 4d2 75. 25x2 81y2 76. 36s2 25t2 373 374 Chapter 5 ■ Polynomials 77. 24x3 10x2 4x Multiply. 78. 12x3 3x 77. 2x(3x 2)(4x 1) 78. 3x(2x 1)(2x 1) 79. 80a 45a 79. 5a(4a 3)(4a 3) 80. 6m(3m 2)(3m 7) 80. 54m3 162m2 84m 81. 3s(5s 2)(4s 1) 82. 7w(2w 3)(2w 3) 81. 60s 39s 6s 83. (x 2)(x 1)(x 3) 84. (y 3)(y 2)(y 4) 82. 28w3 63w 85. (a 1) 86. (x 1)3 3 3 2 3 83. x3 4x2 x 6 84. y3 3y2 10y 24 3 Multiply the following. 2 85. a 3a 3a 1 86. x3 3x2 3x 1 x2 11x 4 87. 3 45 15 x2 29 9 88. x 4 80 20 x 2 2x 2 87. 2 3 3 5 89. [x (y 2)][x (y 2)] x 3 3x 3 88. 3 4 4 5 90. [x (3 y)][x (3 y)] Label the following as true or false. 89. x2 y2 4y 4 90. x2 y2 6y 9 91. (x y)2 x2 y2 92. (x y)2 x2 y2 91. False 93. (x y)2 x2 2xy y2 94. (x y)2 x2 2xy y2 92. False 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. 93. True 94. True 95. 6x2 11x 35 cm2 5 21 96. 3y y in2 2 2 2 97. 10x 3x2 3 98. 100x 2x 99. 25x2 40x 16 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 2x2. 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 5.4 100. 9x2 12x 4 cm2 2 101. x(x 2) or x 2x 102. x(3x 6) or 3x2 6x ■ Multiplying of Polynomials and Special Products 375 100. 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. 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. 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 a2 b2 because. . . . But, wait! Isn’t (a b)2 sometimes equal to a2 b2? What do you think? 105. Is (a b)3 ever equal to a3 b3? Explain. 376 Chapter 5 ■ Polynomials 106. In the following figures, identify the length and the width of the square: a b Length __________ a Width __________ Area __________ b x 2 Length __________ x Width __________ Area __________ 2 107. 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 __________. x If the sides are quadrupled, the area is __________. In general, if the sides are multiplied by n, the area is __________. x 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. x2 36 109. x2 10x 25 For each of the following problems, let x represent the number, then write an expression for the product. 110. x2 8x 16 108. The product of 6 more than a number and 6 less than that number 111. x2 25 109. The square of 5 more than a number 112. 2499 110. The square of 4 less than a number 113. 891 111. The product of 5 less than a number and 5 more than that number 114. 884 115. 9996 Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of the following products. 116. 3575 112. (49)(51) 113. (27)(33) 117. 3584 114. (34)(26) 115. (98)(102) 116. (55)(65) 117. (64)(56)