Section 5.3 Practice Exercises Vocabulary and Key Concepts 1. a. To multiply 2(4x 5), apply the property. b. The conjugate of 4x + 7 is. c. When two conjugates are multiplied the resulting binomial is a difference of. This is given by the formula (a + b)(a b) =. d. When a binomial is squared, the resulting trinomial is a square trinomial. This is given by the formula (a + b) 2 =. Review Exercises 2. Simplify. ( 4x 2 y 2xy + 3xy 2 ) (2x 2 y 4xy 2 ) + (6x 2 y + 5xy) 3. Simplify. ( 2 3x) [5 (6x 2 + 4x + 1)] 4. Given f(x) = 4x 3 5, find the function values. a. f(3) b. f(0) c. f( 2) 5. Given g(x) = x 4 x 2 3, find the function values. a. g( 1) b. g(2) c. g(0) 6. Write the distributive property of multiplication over addition. Give an example of the distributive property. (Answers may vary.) Concept 1: Multiplying Polynomials For Exercises 7 40, multiply the polynomials. (See Examples 1 4.) 7. (7x 4 y)( 6xy 5 ) 8. ( 4a 3 b 7 )( 2ab 3 ) 9. (2.2a 6 b 4 c 7 )(5ab 4 c 3 ) 10. (8.5c 4 d 5 e)(6cd 2 e) 11. 12. 13. 2m 3 n 2 (m 2 n 3 3mn 2 + 4n) Exercise: Multiplying Polynomials PDF Transcript for Exercise: Multiplying Polynomials 14. 3p 2 q (p 3 q 3 pq 2 4p) 15. 16. 17. (x + y)(x 2y) 18. (3a + 5)(a 2) Exercise: Multiplying Binomials PDF Transcript for Exercise: Multiplying Binomials 19. (6x 1)(5 + 2x) 20. (7 + 3x)(x 8) 21. (y 2 12)(2y 2 + 3) 22. (4p 2 1)(2p 2 + 5) 23. (5s + 3t)(5s 2t) 24. (4a + 3b)(4a b) 25. (n 2 + 10)(5n + 3) 26. (m 2 + 8)(3m + 7) 27. (1.3a 4b)(2.5a + 7b) 28. (2.1x 3.5y)(4.7x + 2y) 29. (2x + y)(3x 2 + 2xy + y 2 ) Exercise: Multiplying Polynomials: Binomial Times a Trinomial PDF Transcript for Exercise: Multiplying Polynomials: Binomial Times a Trinomial 30. (h 5k)(h 2 2hk + 3k 2 ) 31. (x 7)(x 2 + 7x + 49) 32. (x + 3)(x 2 3x + 9) 33. (4a b)(a 3 4a 2 b + ab 2 b 3 ) Page 346
34. (3m + 2n)(m 3 + 2m 2 n mn 2 + 2n 3 ) 35. 36. 37. ( x 2 + 2x + 1)(3x 5) 38. (x + y 2z)(5x y + z) 39. 40. Concept 2: Special Case Products: Difference of Squares and Perfect Square Trinomials For Exercises 41 60, multiply by using the special case products. (SeeExample 5.) 41. (a 8)(a + 8) 42. (b + 2)(b 2) 43. (3p + 1)(3p 1) 44. (5q 3)(5q + 3) 45. 46. 47. (3h k)(3h + k) 48. (x 7y)(x + 7y) 49. (3h k) 2 50. (x 7y) 2 51. (t 7) 2 Exercise: Squaring Binomials PDF Transcript for Exercise: Squaring Binomials 52. (w + 9) 2 53. (u + 3v) 2 54. (a 4b) 2 55. 56. 57. (2z 2 w 3 )(2z 2 + w 3 ) 58. (a 4 2b 3 )(a 4 + 2b 3 ) Exercise: Multiply Conjugates PDF Transcript for Exercise: Multiply Conjugates 59. (5x 2 3y) 2 60. (4p 3 2m) 2 For Exercises 61 and 62, the product of two binomials is shown. Determine if the binomials are conjugates. 61. a. ( 5x + 4)(5x + 4) b. ( 5x + 4)(5x 4) 62. a. ( 3 7x)(3 + 7x) b. ( 3 + 7x)(3 + 7x) 63. Multiply the expressions. Explain their similarities. a. (A B)(A + B) b. [(x + y) B][(x + y) + B] 64. Multiply the expressions. Explain their similarities. a. (A + B)(A B) b. [(A + (3h + k)][(a (3h + k)] For Exercises 65 70, multiply the expressions. (See Example 6.) 65. [(w + v) 2][(w + v) + 2]
66. [(x + y) 6][(x + y) + 6] 67. [2 (x + y)][2 + (x + y)] Exercise: Multiplying Conjugates PDF Transcript for Exercise: Multiplying Conjugates 68. [a (b + 1)][a + (b + 1)] 69. [(3a 4) + b][(3a 4) b] 70. [(5p 7) q][(5p 7) + q] 71. Explain how to multiply (x + y) 3. 72. Explain how to multiply (a b) 3. For Exercises 73 76, multiply the expressions. (See Example 7.) 73. (2x + y) 3 74. (x 5y) 3 75. (4a b) 3 76. (3a + 4b) 3 77. Explain how you would multiply the binomials Page 347 78. Explain how you would multiply the binomials For Exercises 79 86, simplify the expressions. 79. 2a 2 (a + 5)(3a + 1) 80. 5y(2y 3)(y + 3) 81. (x + 3)(x 3)(x + 5) 82. (t + 2)(t 3)(t + 1) 83. 3(2x + 7) (4x 1) 2 84. (p + 10) 2 4(p + 6) 2 85. (y + 1) 2 (2y + 3) 2 86. (b 3) 2 (3b 1) 2 Concept 3: Applications Involving a Product of Polynomials For Exercises 87 90, translate from English form to algebraic form. (SeeExample 8.) 87. The square of the sum of r and t 88. The square of a plus the cube of b 89. The difference of x squared and y cubed 90. The square of the product of 3 and a For Exercises 91 94, translate from algebraic form to English form. (SeeExample 8.) 91. p 3 + q 2 92. a 3 b 3 93. xy 2 94. (c + d) 3 95. A rectangular garden has a walk around it of width x. The garden is 20 ft by 15 ft. Write a function representing the combined area A(x) of the garden and walk. Simplify the result. 96. An 8-in. by 10-in. photograph is in a frame of width x. Write a function that represents the area A(x) of the frame alone. Simplify the result. 97. A box is created from a square piece of cardboard 8 in. on a side by cutting a square from each corner and folding up the sides. Let x represent the length of the sides of the squares removed from each corner. (See Example 9.) a. Write a function representing the volume of the box. b. Find the volume if 1-in. squares are removed from the corners.
Section 5.3 Practice Exercises Vocabulary and Key Concepts 1. a. Factoring a polynomial means to write it as a of two or more polynomials. b. The (GCF) of a polynomial is the greatest factor that divides each term of the polynomial evenly. c. The first step toward factoring a polynomial is to factor out the. d. To factor a four-term polynomial, we try the process of factoring by. Review Exercises For Exercises 2 8, perform the indicated operation. 2. ( 4a 3 b 5 c)( 2a 7 c 2 ) 3. (7t 4 + 5t 3 9t) ( 2t 4 + 6t 2 3t) 4. (5x 3 9x + 5) + (4x 3 + 3x 2 2x + 1) (6x 3 3x 2 + x + 1) 5. (5y 2 3)(y 2 + y + 2) 6. (a + 6b) 2 7. 8. Page 365 Concept 1: Factoring Out the Greatest Common Factor For Exercises 9 24, factor out the greatest common factor. (SeeExample 1.) 9. 3x + 12 10. 15x 10 11. 6z 2 + 4z 12. 49y 3 35y 2 13. 4p 6 4p 14. 5q 2 5q 15. 12x 4 36x 2 16. 51w 4 34w 3 17. 9st 2 + 27t 18. 8a 2 b 3 + 12a 2 b 19. 9a 4 b 3 + 27a 3 b 4 18a 2 b 5 20. 3x 5 y 4 15x 4 y 5 + 9x 2 y 7 21. 10x 2 y + 15xy 2 5xy 22. 12c 3 d 15c 2 d + 3cd Exercise: Factoring out the Greatest Common Factor PDF Transcript for Exercise: Factoring out the Greatest Common Factor 23. 13b 2 11a 2 b 12ab 24. 6a 3 2a 2 b + 5a 2 Concept 2: Factoring Out a Negative Factor
For Exercises 25 30, factor out the indicated quantity. (See Example 2.) 25. x 2 10x + 7: Factor out 1. 26. 5y 2 + 10y + 3: Factor out 1. 27. 12x 3 y 6x 2 y 3xy: Factor out 3xy. 28. 32a 4 b 2 + 24a 3 b + 16a 2 b: Factor out 8a 2 b. 29. 2t 3 + 11t 2 3t: Factor out t. 30. 7y 2 z 5yz z: Factor out z. Exercise: Factoring out the Negative Factor PDF Transcript for Exercise: Factoring out the Negative Factor Concept 3: Factoring Out a Binomial Factor For Exercises 31 38, factor out the GCF. (See Example 3.) 31. 2a(3z 2b) 5(3z 2b) 32. 5x(3x + 4) + 2(3x + 4) 33. 2x 2 (2x 3) + (2x 3) 34. z(w 9) + (w 9) 35. y(2x + 1) 2 3(2x + 1) 2 36. a(b 7) 2 + 5(b 7) 2 37. 3y(x 2) 2 + 6(x 2) 2 Exercise: Factoring out a Binomial Factor PDF Transcript for Exercise: Factoring out a Binomial Factor 38. 10z(z + 3) 2 2(z + 3) 2 39. Construct a polynomial that has a greatest common factor of 3x 2. (Answers may vary.) 40. Construct two different trinomials that have a greatest common factor of 5x 2 y 3. (Answers may vary.) 41. Construct a binomial that has a greatest common factor of (c + d). (Answers may vary.) Concept 4: Factoring by Grouping 42. If a polynomial has four terms, what technique would you use to factor it? 43. Factor the polynomials by grouping. a. 2ax ay + 6bx 3by b. 10w 2 5w 6bw + 3b c. Explain why you factored out 3b from the second pair of terms in part (a) but factored out the quantity 3b from the second pair of terms in part (b). 44. Factor the polynomials by grouping. a. 3xy + 2bx + 6by + 4b 2 b. 15ac + 10ab 6bc 4b 2 c. Explain why you factored out 2b from the second pair of terms in part (a) but factored out the quantity 2b from the second pair of terms in part (b). For Exercises 45 64, factor each polynomial by grouping (if possible).(see Examples 4 7.) 45. y 3 + 4y 2 + 3y + 12 46. ab + b + 2a + 2 47. 6p 42 + pq 7q 48. 2t 8 + st 4s 49. 2mx + 2nx + 3my + 3ny 50. 4x 2 + 6xy 2xy 3y 2 Page 366 51. 10ax 15ay 8bx + 12by Exercise: Factor by Grouping PDF Transcript for Exercise: Factor by Grouping 52. 35a 2 15a + 14a 6 53. x 3 x 2 3x + 3 54. 2rs + 4s r 2 55. 6p 2 q + 18pq 30p 2 90p 56. 5s 2 t + 20st 15s 2 60s 57. 100x 3 300x 2 + 200x 600 Exercise: Factoring out the GCF and Factoring by Grouping PDF Transcript for Exercise: Factoring out the GCF and Factoring by Grouping 58. 2x 5 10x 4 + 6x 3 30x 2 59. 6ax by + 2bx 3ay 60. 5pq 12 4q + 15p 61. 4a 3b ab + 12 62. x 2 y + 6x 3x 3 2y 63. 7y 3 21y 2 + 5y 10 64. 5ax + 10bx 2ac + 4bc 65. Explain why the grouping method failed for Exercise 63. 66. Explain why the grouping method failed for Exercise 64. Mixed Exercises 67. Solve the equation U = Av + Acw for A by first factoring out A. 68. Solve the equation S = rt + wt for t by first factoring out t. 69. Solve the equation ay + bx = cy for y. 70. Solve the equation cd + 2x = ac for c. 71. The area of a rectangle of width w is given by A = 2w 2 + w. Factor the right-hand side of the equation to find an expression for the length of the rectangle. 72. The amount in a savings account bearing simple interest at an annual interest rate r for t years is given by A = P + Prt where P is the principal amount invested. a. Solve the equation for P. b. Compute the amount of principal originally invested if the account is worth $12,705 after 3 yr at a 7% interest rate. Page 367
Expanding Your Skills For Exercises 73 80, factor out the greatest common factor and simplify. 73. (a + 3) 4 + 6(a + 3) 5 74. (4 b) 4 2(4 b) 3 75. 24(3x + 5) 3 30(3x + 5) 2 76. 10(2y + 3) 2 + 15(2y + 3) 3 77. (t + 4) 2 (t + 4) 78. (p + 6) 2 (p + 6) 79. 15w 2 (2w 1) 3 + 5w 3 (2w 1) 2 80. 8z 4 (3z 2) 2 + 12z 3 (3z 2) 3