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SECTION 6.3 (e) H- (x + 2 ) 3 + / Special Factoring 337 3 EXERCISE 4 = Factor each polynomial. (a) 1000 + z (b) 81a 6 (c) (x- 3 [(.v + 2 ) + = (x t][(x 2 + 2 + t){x + 2) 2 - (x + + Ax + 4 - xt 2 2)t + l] Sum of cubes 2 - 2t + t ) Multiply. NOW TRY * 3 + 3b 3 3 3) + y /$±CAUTION 3 A c o m m o n error w h e n factoring x 3 3 + y 3 or x —y is to t h i n k that the jry-term has a coefficient o f 2. Since there is n o coefficient o f 2, expressions o f 2 2 the f o r m x + xy + y 2 2 and x — xy + y usually cannot be factored further. The special types o f factoring are summarized here. These should be memorized. 3es of Factorin DiflCr^or^uares Perfect Sqtrar«4Vinoniial NOW TRY ANSWERS 4. (a) (10 + z) • x -y x + 2xy + y = (x + y) x — 2xy + y 2 = (x + y)(x - y) 2 2 2 2 2 lOz + z ) (100 - 2 2 (b) 3(3a + b) • (9a - 3a 6 + b ) (c) (x - 3 + y) • (x - 6x + 9 - xy + 3y + y ) 4 2 2 3 KJf" Sum of Cubes — (JC — y) 2 x — v = (JC — v)(* + xy + v ) Difference of Cubes*?* C * 2 2 3 JC 3 +j = 3 2 (JC + J ) ( J C — 2 2 jcp + y ) 2 2 6.3 EXERCISES WAKN Concept Check © Complete solution available on the Video Resources on DVD / A " " — Work each problem. 1. Which o f the following binomials are differences o f squares? 2 2 A . 64 - k B . 2x 2 - 25 C. k + 9 4 D. 4z - 49 2. Which o f the following binomials are sums or differences o f cubes? 3 6 A . 64 + r 3 B . 125 - p C. 9 x + 125 3 D. (x + y) - 1 3. Which o f the following trinomials are perfect squares? 2 A. x - Sx 4 2 16 B . 4 m + 20m + 25 2 2 C. 9z + 30z + 25 D. 25p - 4 5 / 7 + 81 4. O f the 12 polynomials listed in Exercises 1-3, which ones can be factored by the methods o f this section? 2 5. The binomial 4 x + 64 is an example o f a sum o f two squares that can be factored. Under what conditions can the sum o f two squares be factored? 6. Insert the correct signs in the blanks. (a) 8 + m = (2 — m ) ( 4 — 2m — 3 (b) 3 n - 1 = (n — 2 1)(« — n — Factor each polynomial. See Examples © 2 7. p - 2 16 8. k 2 10. 36m - 25 4 4 13. 64m - 4y 2 m) 1) 1-4. - 2 9 2 11. 18a 4 14. 243x - 9. 25x - 4 98ft 3/ 4 2 2 2 12. 32c - 98d 2 15. (y + z ) - 81 Pnnted by Dorothy Muhammad (dorothy [email protected] j on 11/8/2012 from 67.226 45 111 authorized to use until 4/12/2015. Use beyond the authorized user or valid subscnption date represents a copyright violation. 338 CHAPTER 6 Factoring 2 16. {h + k) 4 19. p 2 - 9 17. 16 - {x + 3y) 4 - 256 2 22. x 20. a + Wx + 25 2 2 2 1 29. x 35. (a - b) ©37. x 2 36. (m - n) 3 50. z 3 3 125p 3 56. 250x 6 + 125 59. m - 9 lOOOx 3 2 1 48. 8 w - 3 125 3 3 51. 6 4 g - 3 3 21h 3 54. 512/ + 27s 3 3 + \6y 57. {y + z ) + 64 125 60. 2 7 r + 1 62. 64 - 729/> 64. Concept Check Consider (x — y) x — 2xy + y — 25 correct? + 343 45. 8x + 1 53. 343p + 125a 3 3 3 3 3 52. 27a - 8b 58. (p - q) - + 2kh + 4 + 4(m - n) + 4 42. r 47. 125x - 216 3 2 s 2 + 64 3 3 2 - h 39. 216 - i 44. 729 + x 3 12r + 9 - - 64 3 55. 24n + 81p 2 2 + 6{x + y) + 9 2 3 46. 2 7 y + . 1 61. 27 - 2 © 41. x 43. 1000 + y 3 - 30. - * 34. (x + y) 38. y - 8y 1 2 32. 80z - 40zw + 5 w - 27 3 27. 4 r 2 3 49. x + 2y - + 8(a - b) + 16 40. 512 - m 3 2 + 2{p + q) + 1 2 3 2 - y - 6k + 9 24. 9>> + 6^z + z 2 2 2t) 2 26. 25c - 20c + 4 - d 31. 98m + 84mn + 1 8 « 33. {p + q) 2 2 - 24a + 16 - b 2 21. k © 23. 4z + 4zw + w 25. 16m - 8m + 1 - n 28. 9 a 2 - 625 2 2 18. 64 - ( r + 6 9 63. 1 2 5 / + z 3 — 25. To factor this polynomial, is the first step 2 RELATING CONCEPTS E X E R C I S E S 65-70 FOR INDIVIDUAL OR GROUP WORK The binomial x — y may be considered either as a difference of squares or a difference of cubes. Work Exercises 65-70 in order. 6 6 65. Factor x — y by first factoring as a difference o f squares. Then factor further by considering one of the factors as a sum o f cubes and the other factor as a difference of cubes. 6 6 66. Based on your answer in Exercise 65, f i l l i n the blank with the correct factors so that x — y is factored completely. 6 6 6 X - y6 = ( X - y)( X + 67. Factor x — y by first factoring as a difference o f cubes. Then factor further by considering one o f the factors as a difference o f squares. 6 6 68. Based on your answer in Exercise 67, f i l l i n the blank with the correct factor so that x — y is factored. 6 6 69. Notice that the factor you wrote in the blank in Exercise 68 is a fourth-degree polynomial, while the two factors you wrote i n the blank in Exercise 66 are both second-degree polynomials. What must be true about the product o f the two factors you wrote in the blank i n Exercise 66? Verify this. 70. I f you have a choice o f factoring as a difference o f squares or a difference of cubes, how should you start to more easily obtain the completely factored form o f the polynomial? Base the answer on your results i n Exercises 65-69. Printed by Dorothy Muhammad ([email protected]) on 11/8/2012 from 67.226.45.111 authorized to use until 4/12/2015. Use beyond the authorized user or valid subscription date represents a copyright violation. 40 CHAPTER R Review of Basic Concepts • FACTORING BY SUBSTITUTION EXAMPLE 7 Factor each polynomial. (a) 6 z 4 - (c) (2a - 2 13z - 5 (b) 10(2a - l) 2 19(2a - 1) - 15 3 l) + 8 Solution 2 2 2 (a) Replace z with u, so u 4 6z - 2 4 = (z ) 2 13z - 5 = 6/r = z. - 1 3 H - 5 Remember to make the L = (2u final substitution. P^*"^ , 1 5) (3M + = (2z' 1) Use F O I L to factor 5) (3?" + 1) Replace i< with r . (Some students prefer to factor this type of trinomial directly using trial and error with F O I L . ) 2 (b) I0(2a - l) 2 = 10H - ~ 5 = ^ ^ — Replace uwith 2 a - 1 . ( 5 U + 19(2a - 19w - 3 ) ( 2 M = [5(2 , 1) - Replace 2a • ' with w Tact or. 5 ) I ) + 3][2(2a ( = (10a - 5 + 3) (4a - (c) 15 15 1) - Let it 5] 2 - 5) 2a - 1. Disiributivc properly = (10a 2) (4a - 7) Add. = 2(5a - l)(4a - 7) Factor out the common i act or. 3 (2a Let 2a - 1 I) + 8 = n + 8 3 = u +2 s Factor, = (u + 2 ) ( « - 2« + 4) 2 = [(2a - 1 - u. Write as a sum o! cubes. 1) + 2] [(2a - I) 2 - 2(2a - 1) + 4] L e t = 2a - L 2 = (2a + 1) (4a - 4a + 1 - 4a + 2 + 4) Add: multiply. = (2a + l)(4a 2 - 8a + 7) Combine like terms. NOW TRY EXERCISES 79, 81, AND 97. < Exercises Factor out the greatest common factor from each polynomial. See Examples 1 and 2 1. 12w + 60 4 4. 9<-. + 81: 3. S i + 24* 6. 5h-j + hj 7. - 4 p V - 2pV y A 9. 4k-m + U 3 2. 15r - 27 5. xy - 5xy- 8. - 3 z V - 3 12. 4(v - 2) + 3( v - 2) 14. (3z + 2) (z + 4) - (z + 6) (z + 4) 13. (5r - 6)(r + 3) - (2r - l)(r + 3) : 35rV 2 11. 2(a + b) + 4m(u + b) 15. 2(m - 1) - 3(m - 1) + 2(m - l ) 18zV 10. 2 8 r V + 7r s - 12ArV 3 3 16. 5(a + 3) - 2(a + 3) + (a + 3) 2 R.4 Factoring Polynomials 41 5 17. Concept Check When directed to completely factor the polynomial 4x~y — Kxy', a student wrote 2xyH2x\< - 4). When the teacher did not give him full credit, he complained because when his answer is multiplied out. the result is the original polynomial. Give the correct answer. 2 Factor each polynomial by grouping. See Example 2. 18. lOafc - 6b + 35a - 21 2 2 19. 6st + 9t - 10s - 15 l 4 20. 15 - 5m - 3 r + m r 2 21. 2m* + 6 - am - 3a 2 2 22. 20c - 8x + 5pz - 2px 2 2 23. p q 2 - 10 - 2a + 5p 2 24. Concept Check Layla factored 16a - 40a - 6a + 15 by grouping and obtained (8a - 3) (2a - 5). Jamal factored the same polynomial and gave an answer of (3 - 8a) (5 - 2a). Which answer is correct? Factor each trinomial, if possible. See Examples 3 and 4. 2 26. 8/i - 2h - 21 27. 3m + 14m + 8 2 29. 15/r + 24p + 8 2 30. 9x + 4.x - 2 25. 6a - 11a + 4 28. 9y - 18y + 8 31. 12a 3 + 10a 2 3 - 42a 2 37. 12,r - xy - y 5 2 32. 36JC + 18x 2 34. 14m + llmr - 15r 2 2 38. 30a + am - m 2 2 33. 6k 2 2 4 2 - 4x 2 2 4 2 42. 16p - 40> + 25 2 2 2 2 45. 4x y + 28rv + 49 46. 9m n + \2mn + 4 47. (a - 3fc) - 6(a - 36) + 9 48. (2p + qf - 10(2p + a) + 25 2 49. Concept Check Column II. Match each polynomial in Column I with its factored form in I II 2 2 A. (x + 5y)(* - 5y) 2 B. (v + 5y) 2 C . (r - 5y) 2 (a) x + \0xy + 25y 2 (b) -V - lOtry + 25v 2 (c) x - 25y 2 2 44. 20p - lOOpa + 125a 2 2 3 39. 24a + 10a fc - 4a 6 41. 9m - 12m + 4 2 6p 36. 12s + list - 5r 2 43. 32a + 48a6 + 186 2 + 5kp 2 35. 5a - lab - 6b 40. 18x + 15.T I - 15xh 2 2 2 2 2 2 (d) 25y - x 50. Concept Check Column II. D. (5y + x) (5y - x) Match each polynomial in Column I with its factored form in I II 3 2 (a) 8x - 27 A. (3 - 2r) (9 + 6x + 4x ) 3 2 (b) 8r + 27 (c) 27 - 8x B. (2x - 3) (4.r + 6x + 9) 3 2 C . (2x + 3) (4x - 6x + 9) Factor each polynomial. See Examples 5 and 6. (In Exercises 53 and 54, factor over the rational numbers.) 2 2 51. 9a - 16 2 52. 16a - 25 53. 36* - — 25 4 2 54. 100v 55. 25s" - 9r 49 57. (a + b) - 16 2 4 60. m - 81 66. 27 3 Z + 729y 2 3 69. 21 - (m + 2nf 4 59. p - 625 3 64. 8m - 27n 3 4 62. r + 27 3 63. 125.r - 27 2 56. 36z - 81v 58. (p - 2a) - 100 61. 8 - a 3 2 3 3 67. (r + 6) - 216 70. 125 - (4a - bf 65. 27y" + 125c 3 6 68. (* + 3) - 27 SECTION 6.5 o Brain Busters 49. 2(x 51. 5(3x - Solving Equations by Factoring 351 Solve each equation. 2 l ) - l(x - 1) - 2 l ) + 3 = -16(3x 2 53. (2x - 3 ) = 2 50. 4 ( 2 * + 3 ) - (2x + 3) - 3 = 0 15 = 0 2 52. 2(x + 3 ) = 5{x + 3) - 2 1) 2 \6x 2 54. 9 x = (5x + 2 ) 2 Solve each problem. See Examples 7 and 8. x+4 2 55. A garden has an area o f 320 f t . Its length is 4 ft more than its width. What are the dimensions o f the garden? (Hint: 320 = 16 • 20) H 56. A square mirror has sides measuring 2 ft less than the sides o f a square painting. I f the difference between their areas is 32 ft , find the lengths o f the sides o f the mirror and the painting. x H 2 © 57. The base o f a parallelogram is 7 ft more than the height. I f the area o f the parallelogram is 60 ft , what are the measures o f the base and the height? 2 58. A sign has the shape o f a triangle. The length o f the base is 3 m less than the height. What are the measures o f the base and the height i f the area is 44 m ? 2 2 59. A farmer has 300 ft o f fencing and wants to enclose a rectangular area o f 5000 f t . What dimensions should she use? (Hint: 5000 = 50 • 100) 2 60. A rectangular landfill has an area o f 30,000 ft . Its length is 200 ft more than its width. What are the dimensions o f the landfill? (Hint: 30,000 = 300 • 100) 61. Find two consecutive integers such that the sum o f their squares is 6 1 . 62. Find two consecutive integers such that their product is 72. 63. A box with no top is to be constructed from a piece o f cardboard whose length measures 6 in. more than its width. The box is to be formed by cutting squares that measure 2 in. on each side from the four corners and then folding up the sides. I f the volume o f the box w i l l be 110 i n . , what are the dimensions o f the piece o f cardboard? 3 w+6 J 2 2 2l_ 2 2 12 2 2r Printed by Dorothy Muhammad [email protected]) on 11/8/2012 from 67.226 45.111 authorized to use until 4/12/2015. Use beyond the authorized user or valid subscription date represents a copyright violation. Solving Quadratic Equations C o m p l e t e the Square Complete the Square to Solve the Equations Worksheet # 4 1 .}Zx'-Ax+ 3 = 0 2.)^+ Z^^-Ax- 32 = 0 4.) 15 = 0 6 . ) ^ - 6 x - 27 = 0 5.)^+ Qn+ Name: 10c-11 = 2 ?_x - 2x- 48 = 0 7 . ) W - 12«+ 32 = 0 8 . ) £ « - 1 2 w + 27 = 9.)3/f- 16»+ 63 = 0 10.) 2^+ 18x+ 65 = http://worksheetplace.com© 2 / Solving Quadratic Equations C o m p l e t e the Square Complete the Square to Solve the Equations Name:. Worksheet # 6 1 .J^x -16x+ 39 = 0 2.) 3 . ) i ^ - 1 6 x + 39 = 0 4.) %yf- 14/i+ 5.)y^-6x- 6.)%rf- 2 55 = 0 2 2 3 x - 14x+ 13 = 0 10«- $x*+4x-?.\ 33 = 0 11=0 7.)^JC +8X-9 = 0 8.) 9.)^e- 10x+21 10.)3« + 18«+ 56 = =0 http://worksheetplace.com© 2 =0