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3.4 Multiplying Polynomials 3.4 OBJECTIVES 1. Find the product of a monomial and a polynomial 2. Find the product of two polynomials You have already had some experience in multiplying polynomials. In Section 1.7 we stated the first property of exponents and used that property to find the product of two monomial terms. Let’s review briefly. Step by Step: NOTE The first property of Step 1 Step 2 To Find the Product of Monomials Multiply the coefficients. Use the first property of exponents to combine the variables. exponents: xm xn xmn Example 1 Multiplying Monomials Multiply 3x2y and 2x3y5. Write (3x2y)(2x3y5) (3 2)(x2 x3)(y y5) NOTE Once again we have used the commutative and associative properties to rewrite the problem. Multiply Add the exponents. the coefficients. 6x5y6 CHECK YOURSELF 1 Multiply. (a) (5a2b)(3a2b4) © 2001 McGraw-Hill Companies NOTE You might want to review Section 1.2 before going on. (b) (3xy)(4x3y5) 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 1.2. That property leads us to the following rule for multiplication. Rules and Properties: To Multiply a Polynomial by a Monomial NOTE Distributive property: a(b c) ab ac Use the distributive property to multiply each term of the polynomial by the monomial. Example 2 Multiplying a Monomial and a Binomial (a) Multiply 2x 3 by x. 281 282 CHAPTER 3 POLYNOMIALS Write NOTE With practice you will do this step mentally. x(2x 3) x 2x x 3 2x2 3x Multiply x by 2x and then by 3, the terms of the polynomial. That is, “distribute” the multiplication over the sum. (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 all the steps of the process. With practice you can write the product directly, and you should try to do so. (b) 5y2(2y3 4) 5y2 2y3 5y2 4 10y5 20y2 (c) 5c(4c2 8c) (5c) (4c2) (5c) (8c) 20c3 40c2 (d) 3c2d 2(7cd 2 5c2d 3) 3c2d2 7cd 2 3c2d2 5c2d 3 21c3d 4 15c4d 5 CHECK YOURSELF 3 Multiply. (a) 3(5a2 2a 7) (c) 5m(8m2 5m) Example 4 Multiplying Binomials (a) Multiply x 2 by x 3. (b) 4x2(8x3 6) (d) 9a2b(3a3b 6a2b4) © 2001 McGraw-Hill Companies NOTE Again we have shown MULTIPLYING POLYNOMIALS that each term, x and 2, of the first binomial is multiplied by each term, x and 3, of the second binomial. 283 We can think of x 2 as a single quantity and apply the distributive property. NOTE Note that this ensures SECTION 3.4 (x 2)(x 3) 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. (Think of a 3 as a single quantity and distribute.) (a 3)(a 4) (a 3)a (a 3)(4) a a 3 a [(a 4) (3 4)] a2 3a (4a 12) a 3a 4a 12 2 Note that the parentheses are needed here because a minus sign precedes the binomial. a2 7a 12 CHECK YOURSELF 4 Multiply. (a) (x 4)(x 5) (b) (y 5)(y 6) 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): 1. (x 2)(x 3) NOTE Remember this by F! xx Find the product of the first terms of the factors. 2. (x 2)(x 3) NOTE Remember this by O! Find the product of the outer terms. x3 3. (x 2)(x 3) NOTE Remember this by I! 2x Find the product of the inner terms. © 2001 McGraw-Hill Companies 4. (x 2)(x 3) NOTE Remember this by L! 23 Find the product of the last terms. Combining the four steps, we have NOTE Of course these are the same four terms found in Example 4a. (x 2)(x 3) x2 3x 2x 6 NOTE It’s called FOIL to give x2 5x 6 you an easy way of remembering the steps: First, Outer, Inner, and Last. With practice, the FOIL method will let you write the products quickly and easily. Consider Example 5, which illustrates this approach. 284 CHAPTER 3 POLYNOMIALS Example 5 Using the FOIL Method Find the following products, using the FOIL method. F x x L 4 5 (a) (x 4)(x 5) 4x I 5x O should combine the outer and inner products mentally and write just the final product. x2 5x 4x 20 F O I L x 9x 20 2 F x x L (7)(3) (b) (x 7)(x 3) 7x I Combine the outer and inner products as 4x. 3x O x2 4x 21 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 coefficients other than 1 or with more than one variable. Example 6 Using the FOIL Method Find the following products, using the FOIL method. F 12x2 L 6 (a) (4x 3)(3x 2) 9x I 8x O 12x2 x 6 Combine: 9x 8x x © 2001 McGraw-Hill Companies NOTE When possible, you MULTIPLYING POLYNOMIALS 6x2 SECTION 3.4 285 35y2 (b) (3x 5y)(2x 7y) 10xy Combine: 10xy 21xy 31xy 21xy 6x 31xy 35y2 2 The following rule summarizes our work in multiplying binomials. Step by Step: Step 1 Step 2 Step 3 To Multiply Two Binomials 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 Multiply. (a) (5x 2)(3x 7) (b) (4a 3b)(5a 4b) (c) (3m 5n)(2m 3n) Sometimes, especially with larger polynomials, it is easier to use the vertical method to find their product. This is the same method you originally learned when multiplying two large integers. Example 7 Multiplying Using the Vertical Method Use the vertical method to find the product of (3x 2)(4x 1). First, we rewrite the multiplication in vertical form. 3x 2 4x (1) Multiplying the quantity 3x 2 by 1 yields © 2001 McGraw-Hill Companies 3x 2 4x (1) 3x (2) Note that we maintained the columns of the original binomial when we found the product. We will continue with those columns as we multiply by the 4x term. 3x 2 4x (1) 3x (2) 12x2 8x 12x2 5x (2) 286 CHAPTER 3 POLYNOMIALS We could write the product as (3x 2)(4x 1) 12x2 5x 2. CHECK YOURSELF 7 Use the vertical method to find the product of (5x 3)(2x 1). We’ll use the vertical method again in our next example. This time, we will multiply a binomial and a trinomial. Note that the FOIL method can never work for anything but the product of two binomials. Example 8 Using the Vertical Method Multiply x2 5x 8 by x 3. Step 1 x2 5x 8 x 3 3x2 15x 24 x2 5x 8 x 3 Step 2 3x2 15x 24 x 5x2 8x 3 Note that this line is shifted over so that like terms are in the same columns. x2 5x 8 x 3 Step 3 method ensures that each term of one factor multiplies each term of the other. That’s why it works! Now multiply each term by x. 3x2 15x 24 x 5x2 8x 3 x 2x 7x 24 3 2 Now add to combine like terms to write the product. CHECK YOURSELF 8 Multiply 2x2 5x 3 by 3x 4. CHECK YOURSELF ANSWERS 1. 3. 4. 5. 6. 7. (a) 15a4b5; (b) 12x4y6 2. (a) 2y3 6y2; (b) 6w5 15w3 (a) 15a2 6a 21; (b) 32x5 24x2; (c) 40m3 25m2; (d) 27a5b2 54a4b5 (a) x2 9x 20; (b) y2 y 30 (a) x2 13x 42; (b) x2 2x 15; (c) x2 10x 16 (a) 15x2 29x 14; (b) 20a2 31ab 12b2; (c) 6m2 19mn 15n2 10x2 x 3 8. 6x3 7x2 11x 12 © 2001 McGraw-Hill Companies NOTE Using this vertical Multiply each term of x2 5x 8 by 3. Name 3.4 Exercises Section Date Multiply. 1. (5x2)(3x3) ANSWERS 2. (7a5)(4a6) 1. 3. (2b2)(14b8) 4. (14y4)(4y6) 2. 3. 4. 6 7 5. (10p )(4p ) 8 7 6. (6m )(9m ) 5. 6. 7. (4m5)(3m) 8. (5r7)(3r) 7. 8. 9. (4x3y2)(8x2y) 10. (3r4s2)(7r2s5) 9. 10. 11. (3m5n2)(2m4n) 12. (7a3b5)(6a4b) 11. 12. 13. 13. 5(2x 6) 14. 4(7b 5) 14. 15. 15. 3a(4a 5) 16. 5x(2x 7) 16. 17. 17. 3s2(4s2 7s) 18. 9a2(3a3 5a) 18. © 2001 McGraw-Hill Companies 19. 19. 2x(4x2 2x 1) 20. 5m(4m3 3m2 2) 20. 21. 22. 21. 3xy(2x y xy 5xy) 2 2 22. 5ab (ab 3a 5b) 2 23. 23. 6m2n(3m2n 2mn mn2) 24. 8pq2(2pq 3p 5q) 24. 287 ANSWERS 25. Multiply. 26. 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) 43. (x 5)2 44. (y 8)2 45. (y 9)2 46. (2a 3)2 47. (6m n)2 48. (7b c)2 49. (a 5)(a 5) 50. (x 7)(x 7) 51. (x 2y)(x 2y) 52. (7x y)(7x y) 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 48. 49. 50. 51. 52. 288 © 2001 McGraw-Hill Companies 47. ANSWERS 53. (5s 3t)(5s 3t) 54. (9c 4d)(9c 4d) 53. 54. 55. Multiply, using the vertical method. 56. 55. (x 2)(3x 5) 56. (a 3)(2a 7) 57. 58. 57. (2m 5)(3m 7) 58. (5p 3)(4p 1) 59. 60. 59. (3x 4y)(5x 2y) 60. (7a 2b)(2a 4b) 61. 62. 61. (a2 3ab b2)(a2 5ab b2) 62. (m2 5mn 3n2)(m2 4mn 2n2) 63. 64. 63. (x 2y)(x2 2xy 4y2) 64. (m 3n)(m2 3mn 9n2) 65. 66. 65. (3a 4b)(9a2 12ab 16b2) 66. (2r 3s)(4r2 6rs 9s2) 67. 68. 69. Multiply. 70. 67. 2x(3x 2)(4x 1) 68. 3x(2x 1)(2x 1) 71. 72. 69. 5a(4a 3)(4a 3) 70. 6m(3m 2)(3m 7) 73. © 2001 McGraw-Hill Companies 74. 71. 3s(5s 2)(4s 1) 72. 7w(2w 3)(2w 3) 75. 76. 73. (x 2)(x 1)(x 3) 74. (y 3)(y 2)(y 4) 75. (a 1)3 76. (x 1)3 289 ANSWERS Multiply the following. 77. 78. 79. 77. 2 3 3 5 78. 3 4 4 5 x 2 2x 2 80. 81. x 3 3x 3 82. 79. [x (y 2)][x (y 2)] 83. 84. 80. [x (3 y)][x (3 y)] 85. 86. Label the following as true or false. 87. 81. (x y)2 x2 y2 88. 82. (x y)2 x2 y2 83. (x y)2 x2 2xy y2 84. (x y)2 x2 2xy y2 85. 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. 86. Area. The base of a triangle measures 3y 7 inches (in.) and the height is 87. Revenue. The price of an item is given by p 2x 10. 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. 88. Revenue. The price of an item is given by p 2x2 100. Find the polynomial that represents the revenue generated from the sale of x items. 290 © 2001 McGraw-Hill Companies 2y 3 in. Express the area of the triangle in terms of y. ANSWERS 89. 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. 89. 90. 30 in. x 20 in. a. 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. b. c. d. Length of Side of Corner Square 1 in. 2 in. 3 in. Length of Box Width of Box Depth of Box Volume of Box e. f. g. h. n in. Write a general formula 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 ______? 90. (a) Multiply (x 1) (x 1) © 2001 McGraw-Hill Companies (b) Multiply (x 1)(x2 x 1) (c) Multiply (x 1)(x3 x2 x 1) (d) Based on your results to (a), (b), and (c), find the product (x 1) (x29 x28 x 1). Getting Ready for Section 3.5 [Section 1.4] Simplify. (a) (c) (e) (g) (3a)(3a) (5x)(5x) (2w)(2w) (4r)(4r) (b) (d) (f) (h) (3a)2 (5x)2 (2w)2 (4r)2 291 Answers 1. 15x5 3. 28b10 5. 40p13 7. 12m6 9. 32x5y3 11. 6m9n3 2 4 3 3 13. 10x 30 15. 12a 15a 17. 12s 21s 19. 8x 4x2 2x 3 2 2 3 2 2 4 2 3 2 3 3 21. 6x y 3x y 15x y 23. 18m n 12m n 6m n 25. x2 5x 6 2 2 2 27. m 14m 45 29. p p 56 31. w 30w 200 33. 3x2 29x 40 35. 6x2 x 12 37. 12a2 31ab 9b2 2 2 2 39. 21p 13pq 20q 41. 6x 23xy 20y2 43. x2 10x 25 2 2 2 2 45. y 18y 81 47. 36m 12mn n 49. a 25 51. x2 4y2 2 2 2 2 53. 25s 9t 55. 3x 11x 10 57. 6m m 35 59. 15x2 14xy 8y2 61. a4 2a3b 15a2b2 8ab3 b4 63. x3 8y3 65. 27a3 64b3 67. 24x3 10x2 4x 3 3 2 69. 80a 45a 71. 60s 39s 6s 73. x3 4x2 x 6 75. a3 3a2 3a 1 81. False 89. h. 16r2 © 2001 McGraw-Hill Companies g. 16r2 83. True a. 9a2 x2 11x 4 79. x2 y2 4y 4 3 45 15 85. 6x2 11x 35cm2 87. 2x2 10x 2 2 2 b. 9a c. 25x d. 25x e. 4w2 f. 4w2 77. 292 3.5 Special Products 3.5 OBJECTIVES 1. Square a binomial 2. Find the product of two binomials that differ only in their sign 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) x2 2xy y 2 The patterns above lead us to the following rule. Step by Step: Step 1 Step 2 Step 3 To Square a Binomial 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. Example 1 Squaring a Binomial (a) (x 3)2 x2 2 · x · 3 32 CA UTI O N 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 © 2001 McGraw-Hill Companies (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 (d) (5c 3d )2 (5c)2 2(5c)(3d) (3d)2 25c2 30cd 9d 2 Again we have shown all the steps. With practice you can write just the square. 293 294 CHAPTER 3 POLYNOMIALS CHECK YOURSELF 1 Multiply. (a) (2x 1)2 (b) (4x 3y)2 Example 2 Squaring a Binomial Find ( y 4)2. 2 2 ( y 4)2 is not equal to y2 42 or y2 16 The correct square is ( y 4)2 y2 8y 16 The middle term is twice the product of y and 4. CHECK YOURSELF 2 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 terms differ only in sign. Let’s see what happens when we multiply. (x y)(x y) x2 xy xy y2 x2 y2 0 Because the middle term becomes 0, we have the following rule. Rules and Properties: 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. © 2001 McGraw-Hill Companies (2 3) 2 3 because 52 4 9 2 NOTE You should see that SPECIAL PRODUCTS SECTION 3.5 295 Let’s look at the application of this rule in Example 3. Example 3 Multiplying Polynomials Multiply each pair of binomials. (a) (x 5)(x 5) x2 52 Square of the first term Square of the second term x2 25 NOTE (b) (x 2y)(x 2y) x2 (2y)2 (2y) (2y)(2y) 2 4y 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 3 Find the products. (a) (a 6)(a 6) (c) (5n 2p)(5n 2p) (b) (x 3y)(x 3y) (d) (7b 3c)(7b 3c) © 2001 McGraw-Hill Companies 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. Example 4 Multiplying Polynomials (a) x (x 3)(x 3) x(x2 9) x3 9x These binomials differ only in the sign. CHAPTER 3 POLYNOMIALS (b) (x 1) (x 5)(x 5) (x 1)(x2 25) These binomials differ only in the sign. With two binomials, use the FOIL method. x3 x2 25x 25 (c) (2x 1) (x 3) (2x 1) (x 3)(2x 1)(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)(4x2 1) 4x3 12x2 x 3 CHECK YOURSELF 4 Multiply. (a) 3x(x 5)(x 5) (c) (x 7)(3x 1)(x 7) (b) (x 4)(2x 3)(2x 3) CHECK YOURSELF ANSWERS 1. 2. 3. 4. (a) 4x 2 4x 1; (b) 16x 2 24xy 9y 2 (a) x 2 10x 25; (b) 9a 2 12a 4; (c) y 2 14y 49; (d) 25x 2 20xy 4y 2 (a) a 2 36; (b) x 2 9y 2; (c) 25n 2 4p 2; (d) 49b 2 9c 2 (a) 3x 3 75x; (b) 4x 3 16x 2 9x 36; (c) 3x 3 x 2 147x 49 © 2001 McGraw-Hill Companies 296 Name 3.5 Exercises Section Date Find each of the following squares. ANSWERS 1. (x 5)2 2. (y 9)2 1. 3. (w 6)2 2. 4. (a 8)2 3. 5. (z 12)2 6. ( p 20)2 7. (2a 1) 8. (3x 2) 4. 5. 2 2 9. (6m 1)2 6. 7. 10. (7b 2)2 8. 11. (3x y)2 9. 12. (5m n)2 10. 13. (2r 5s)2 14. (3a 4b)2 15. (8a 9b) 16. (7p 6q) 11. 12. 2 2 13. 14. 1 17. x 2 2 1 18. w 4 2 15. 16. 17. Find each of the following products. 19. (x 6)(x 6) 20. ( y 8)( y 8) 18. © 2001 McGraw-Hill Companies 19. 21. (m 12)(m 12) 22. (w 10)(w 10) 20. 21. 1 23. x 2 1 x 2 2 24. x 3 2 x 3 22. 23. 24. 297 ANSWERS 25. 25. (p 0.4)(p 0.4) 26. (m 0.6)(m 0.6) 27. (a 3b)(a 3b) 28. (p 4q)(p 4q) 29. (4r s)(4r s) 30. (7x y)(7x y) 31. (8w 5z)(8w 5z) 32. (7c 2d )(7c 2d) 33. (5x 9y)(5x 9y) 34. (6s 5t)(6s 5t) 35. x(x 2)(x 2) 36. a(a 5)(a 5) 37. 2s(s 3r)(s 3r) 38. 5w(2w z)(2w z) 39. 5r(r 3)2 40. 3x(x 2)2 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. For each of the following problems, let x represent the number, then write an expression for the product. 39. 40. 41. The product of 6 more than a number and 6 less than that number 41. 42. The square of 5 more than a number 42. 43. 43. The square of 4 less than a number 44. 45. 44. The product of 5 less than a number and 5 more than that number 46. 48. 49. 45. (49)(51) 46. (27)(33) 47. (34)(26) 48. (98)(102) 49. (55)(65) 50. (64)(56) 50. 298 © 2001 McGraw-Hill Companies Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of the following products. 47. ANSWERS 51. 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? 51. 52. 53. 54. 52. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the 55. area of the square as a polynomial. 3x 2 cm 3x 2 cm 53. 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? 54. Is (a b)3 ever equal to a3 b3? Explain. 55. In the following figures, identify the length, width, and area of the square: a b Length a Width b Area a 3 a Width Area © 2001 McGraw-Hill Companies 3 x x Length x2 2x Length 2x Width Area 299 ANSWERS 56. The square below is x units on a side. The area is 56. . a. Draw a picture of what happens when the sides are doubled. The area is . b. Continue the picture to show what happens when the sides are tripled. The area is . c. If the sides are quadrupled, the area is d. In general, if the sides are multiplied by n, the area is . e. If each side is increased by 3, the area is increased by . f. If each side is decreased by 2, the area is decreased by g. In general, if each side is increased by n, the area is increased by each side is decreased by n, the area is decreased by . . . , and if h. x x Getting Ready for Section 3.6 [Section 1.7] Divide. (a) 2x2 2x (b) 3a3 3a (c) 6p3 2p2 (d) 10m4 5m2 (e) 20a3 5a3 (f) 6x2y 3xy (g) 12r3s2 4rs (h) 49c4d 6 7cd3 Answers 3. w2 12w 36 11. 9x2 6xy y2 15. 64a2 144ab 81b2 19. x2 36 21. m2 144 1 25. p2 0.16 29. 16r2 s2 4 64w2 25z2 33. 25x2 81y2 35. x3 4x 37. 2s3 18r2s 3 2 2 2 5r 30r 45r 41. x 36 43. x 8x 16 45. 2499 884 49. 3575 51. 25x2 40x 16 55. a. x b. a2 c. 3p d. 2m2 f. 2x 300 1 4 2 27. a 9b2 17. x2 x 23. x2 31. 39. 47. 53. 5. z2 24z 144 7. 4a2 4a 1 2 13. 4r 20rs 25s2 g. 3r2s h. 7c3d 3 e. 4 © 2001 McGraw-Hill Companies 1. x2 10x 25 9. 36m2 12m 1 3.6 Dividing Polynomials 3.6 OBJECTIVES 1. Find the quotient when a polynomial is divided by a monomial 2. Find the quotient of two polynomials In Section 1.7, we introduced the second property of exponents, which was used to divide one monomial by another monomial. Let’s review that process. Step by Step: Step 1 Step 2 xm xmn xn Divide the coefficients. Use the second property of exponents to combine the variables. Example 1 NOTE The second property says: If x is not zero, To Divide a Monomial by a Monomial Dividing Monomials Divide: 8 4 2 8x4 4x42 2x2 (a) Subtract the exponents. 4x2 (b) 45a5b3 5a3b2 9a2b CHECK YOURSELF 1 Divide. (a) 16a5 8a3 (b) 28m4n3 7m3n © 2001 McGraw-Hill Companies Now let’s look at how this can be extended to divide any polynomial by a monomial. For example, to divide 12a3 8a2 by 4a, proceed as follows: NOTE Technically, this step depends on the distributive property and the definition of division. 12a3 8a2 12a3 8a2 4a 4a 4a Divide each term in the numerator by the denominator, 4a. Now do each division. 3a2 2a 301 302 CHAPTER 3 POLYNOMIALS The work above leads us to the following rule. Step by Step: To Divide a Polynomial by a Monomial 1. Divide each term of the polynomial by the monomial. 2. Simplify the results. Example 2 Dividing by Monomials Divide each term by 2. (a) 4a2 8 4a2 8 2 2 2 2a2 4 Divide each term by 6y. (b) 24y3 (18y2) 24y3 18y2 6y 6y 6y 4y2 3y Remember the rules for signs in division. (c) 15x2 10x 15x2 10x 5x 5x 5x 3x 2 NOTE With practice you can (d) 14x4 28x3 21x2 14x4 28x3 21x2 7x2 7x2 7x2 7x2 2x2 4x 3 (e) 9a3b4 6a2b3 12ab4 9a3b4 6a2b3 12ab4 3ab 3ab 3ab 3ab 3a2b3 2ab2 4b3 CHECK YOURSELF 2 Divide. (a) 20y3 15y2 5y (c) 16m4n3 12m3n2 8mn 4mn (b) 8a3 12a2 4a 4a © 2001 McGraw-Hill Companies write just the quotient. DIVIDING POLYNOMIALS SECTION 3.6 303 We are now ready to look at dividing one polynomial by another polynomial (with more than one term). The process is very much like long division in arithmetic, as Example 3 illustrates. Example 3 Dividing by Binomials Divide x2 7x 10 by x 2. NOTE The first term in the Step 1 dividend, x2, is divided by the first term in the divisor, x. Step 2 x x 2B x2 7x 10 Divide x2 by x to get x. x x 2Bx2 7x 10 x2 2x Multiply the divisor, x 2, by x. REMEMBER To subtract Step 3 x2 2x, mentally change each sign to x2 2x, and add. Take your time and be careful here. It’s where most errors are made. x x 2Bx2 7x 10 x2 2x 5x 10 Subtract and bring down 10. Step 4 x 5 x 2Bx2 7x 10 x2 2x 5x 10 NOTE Notice that we repeat the process until the degree of the remainder is less than that of the divisor or until there is no remainder. Step 5 Divide 5x by x to get 5. x 5 x 2Bx2 7x 10 x2 2x 5x 10 5x 10 0 © 2001 McGraw-Hill Companies Multiply x 2 by 5 and then subtract. The quotient is x 5. CHECK YOURSELF 3 Divide x2 9x 20 by x 4. In Example 3, we showed all the steps separately to help you see the process. In practice, the work can be shortened. 304 CHAPTER 3 POLYNOMIALS Example 4 Dividing by Binomials Divide x2 x 12 by x 3. Step 1 Divide x2 by x to get x, the first term of the quotient. Step 2 Multiply x 3 by x. Step 3 Subtract and bring down 12. Remember to mentally change the signs to x2 3x and add. Step 4 Divide 4x by x to get 4, the second term of the quotient. Step 5 Multiply x 3 by 4 and subtract. 4x 12 4x 12 0 The quotient is x 4. CHECK YOURSELF 4 Divide. (x2 2x 24) (x 4) You may have a remainder in algebraic long division just as in arithmetic. Consider Example 5. Example 5 Dividing by Binomials Divide 4x2 8x 11 by 2x 3. 2x 1 2x 3B4x2 8x 11 4x2 6x Quotient 2x 11 2x 3 Divisor 8 Remainder This result can be written as 4x2 8x 11 2x 3 2x 1 8 2x 3 Quotient CHECK YOURSELF 5 Divide. (6x2 7x 15) (3x 5) Remainder Divisor © 2001 McGraw-Hill Companies out a problem like 408 17, to compare the steps. x 4 x 3Bx x 12 x2 3x 2 NOTE You might want to write DIVIDING POLYNOMIALS SECTION 3.6 305 The division process shown in our previous examples can be extended to dividends of a higher degree. The steps involved in the division process are exactly the same, as Example 6 illustrates. Example 6 Dividing by Binomials Divide 6x3 x2 4x 5 by 3x 1. 2x2 x 1 3x 1B6x x2 4x 5 6x3 2x2 3 3x2 4x 3x2 x 3x 5 3x 1 6 The result can be written as 6 6x3 x2 4x 5 2x2 x 1 3x 1 3x 1 CHECK YOURSELF 6 Divide 4x3 2x2 2x 15 by 2x 3. Suppose that the dividend is “missing” a term in some power of the variable. You can use 0 as the coefficient for the missing term. Consider Example 7. Example 7 Dividing by Binomials Divide x3 2x2 5 by x 3. x2 5x 15 x 3Bx 2x2 0x 5 x3 3x2 © 2001 McGraw-Hill Companies 3 5x2 0x 5x2 15x Write 0x for the “missing” term in x. 15x 5 15x 45 40 This result can be written as 40 x3 2x2 5 x2 5x 15 x3 x3 CHAPTER 3 POLYNOMIALS CHECK YOURSELF 7 Divide. (4x3 x 10) (2x 1) You should always arrange the terms of the divisor and dividend in descending-exponent form before starting the long division process, as illustrated in Example 8. Example 8 Dividing by Binomials Divide 5x2 x x3 5 by 1 x2. Write the divisor as x2 1 and the dividend as x3 5x2 x 5. x5 x2 1Bx3 5x2 x 5 x3 x 5x2 5x2 5 5 Write x3 x, the product of x and x2 1, so that like terms fall in the same columns. 0 CHECK YOURSELF 8 Divide: (5x2 10 2x3 4x) (2 x2) CHECK YOURSELF ANSWERS 2. (a) 4y2 3y; (b) 2a2 3a 1; (c) 4m3n2 3m2n 2 20 6 3. x 5 4. x 6 5. 2x 1 6. 2x2 4x 7 3x 5 2x 3 11 7. 2x2 x 1 8. 2x 5 2x 1 1. (a) 2a2; (b) 4mn2 © 2001 McGraw-Hill Companies 306 Name Exercises 3.6 Section Date Divide. 1. 18x6 9x2 2. 20a7 5a5 ANSWERS 1. 3. 35m3n2 7mn2 4. 42x5y2 6x3y 2. 3. 5. 3a 6 3 9b2 12 7. 3 16a3 24a2 9. 4a 4x 8 4 4. 10m2 5m 8. 5 6. 6. 5. 7. 8. 9x3 12x2 10. 3x 9. 10. 12m 6m 3m 2 11. 20b 25b 5b 3 12. 2 11. 12. 13. 18a4 12a3 6a2 6a 14. 21x5 28x4 14x3 7x 13. 14. 15. 20x4y2 15x2y3 10x3y 5x2y 16. 16m3n3 24m2n2 40mn3 8mn2 15. 16. 17. Perform the indicated divisions. 17. x2 5x 6 x2 18. x2 8x 15 x3 18. © 2001 McGraw-Hill Companies 19. 19. x2 x 20 x4 2x2 5x 3 21. 2x 1 23. 2x2 3x 5 x3 20. x2 2x 35 x5 20. 21. 3x2 20x 32 22. 3x 4 22. 3x2 17x 12 x6 24. 24. 23. 307 ANSWERS 25. 25. 4x2 18x 15 x5 26. 3x2 18x 32 x8 27. 6x2 x 10 3x 5 28. 4x2 6x 25 2x 7 29. x3 x2 4x 4 x2 30. x3 2x2 4x 21 x3 31. 4x3 7x2 10x 5 4x 1 32. 2x3 3x2 4x 4 2x 1 33. x3 x2 5 x2 34. x3 4x 3 x3 35. 25x3 x 5x 2 36. 8x3 6x2 2x 4x 1 37. 2x2 8 3x x3 x2 38. x2 18x 2x3 32 x4 39. x4 1 x1 40. x4 x2 16 x2 41. x3 3x2 x 3 x2 1 42. x3 2x2 3x 6 x2 3 43. x4 2x2 2 x2 3 44. x4 x2 5 x2 2 45. y3 1 y1 46. y3 8 y2 47. x4 1 x2 1 48. x6 1 x3 1 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 43. 44. 45. 46. 47. 48. 308 © 2001 McGraw-Hill Companies 42. ANSWERS 49. y2 y c y2 49. Find the value of c so that y1 50. Find the value of c so that 50. x3 x2 x c x1 x2 1 51. 51. Write a summary of your work with polynomials. Explain how a polynomial is recognized, and explain the rules for the arithmetic of polynomials—how to add, subtract, multiply, and divide. What parts of this chapter do you feel you understand very well, and what part(s) do you still have questions about, or feel unsure of? Exchange papers with another student and compare your questions. 52. 53. (a) (b) (c) 52. A funny (and useful) thing about division of polynomials: To find out about this (d) funny thing, do this division. Compare your answer with another student’s. 54. (x 2)B2x2 3x 5 Is there a remainder? (a) (b) Now, evaluate the polynomial 2x2 3x 5 when x 2. Is this value the same as the remainder? (c) Try (x 3)B5x2 2x 1 Is there a remainder? Evaluate the polynomial 5x2 2x 1 when x 3. Is this value the same as the remainder? What happens when there is no remainder? Try (x 6)B3x3 14x2 23x 6 (d) Is the remainder zero? Evaluate the polynomial 3x3 14x 23x 6 when x 6. Is this value zero? Write a description of the patterns you see. When does the pattern hold? Make up several more examples, and test your conjecture. 53. (a) Divide x2 1 x1 (b) Divide x3 1 x1 © 2001 McGraw-Hill Companies (d) Based on your results to (a), (b), and (c), predict 54. (a) Divide (c) Divide x2 x 1 x1 (c) Divide x50 1 x1 (b) Divide x4 x3 x2 x 1 x1 (d) Based on your results to (a), (b), and (c), predict x4 1 x1 x3 x2 x 1 x1 x10 x9 x8 x 1 x1 309 Answers 1. 2x4 3. 5m2 3 13. 3a 2a2 a 21. x 3 5. a 2 7. 3b2 4 9. 4a2 6a 11. 4m 2 2 2 15. 4x y 3y 2x 17. x 3 19. x 5 23. 2x 3 4 x3 25. 4x 2 5 x5 5 8 31. x2 2x 3 29. x2 x 2 3x 5 4x 1 9 2 33. x2 x 2 35. 5x2 2x 1 x2 5x 2 2 2 41. x 3 37. x 4x 5 39. x3 x2 x 1 x2 1 43. x2 1 2 45. y2 y 1 47. x2 1 49. c 2 x 3 2 3 2 51. 53. (a) x 1; (b) x x 1; (c) x x x 1; 27. 2x 3 © 2001 McGraw-Hill Companies (d) x49 x48 x 1 310