5.3 MULTIPLICATION OF BINOMIALS
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1 238 (5 16) Chpter 5 Polynomils nd Exponents 5.3 MULTIPLICATION OF BINOMIALS In this section The FOIL Method Multiplying Binomils Quickly In Section 5.2 you lerned to multiply polynomils. In this section you will lern rule tht mkes multipliction of inomils simpler. The FOIL Method We cn use the distriutive property to find the product of two inomils. For exmple, (x 2)(x 3) (x 2)x (x 2)3 Distriutive property x 2 2x 3x 6 Distriutive property x 2 5x 6 Comine like terms. There re four terms in x 2 2x 3x 6. The term x 2 is the product of the first term of ech inomil, x nd x. The term 3x is the product of the two outer terms, 3 nd x. The term 2x is the product of the two inner terms, 2 nd x. The term 6 is the product of the lst term of ech inomil, 2 nd 3. We cn connect the terms multiplied y lines s follows: F L (x 2)(x 3) I O F First terms O Outer terms I Inner terms L Lst terms If you rememer the word FOIL, you cn get the product of the two inomils much fster thn writing out ll of the steps ove. This method is clled the FOIL method. The nme should mke it esier to rememer. E X A M P L E 1 helpful hint You my hve to prctice FOIL while to get good t it. However, the etter you re t FOIL, the esier you will find fctoring in Chpter 6. Using the FOIL method Find ech product. ) (x 2)(x 4) ) (2x 5)(3x 4) c) ( )(2 ) d) (x 3)(y 5) F F O I L ) (x 2)(x 4) x 2 4x 2x 8 x 2 2x 8 I O L Comine the like terms. ) (2x 5)(3x 4) 6x 2 8x 15x 20 6x 2 7x 20 Comine the like terms. c) ( )(2 ) d) (x 3)(y 5) xy 5x 3y 15 There re no like terms to comine.
2 5.3 Multipliction of Binomils (5 17) 239 FOIL cn e used to multiply ny two inomils. The inomils in the next exmple hve higher powers thn those of Exmple 1. E X A M P L E 2 study tip Rememer tht everything we do in solving prolems is sed on principles (which re lso clled rules, theorems, nd definitions). These principles justify the steps we tke. Be sure tht you understnd the resons. If you just memorize procedures without understnding, you will soon forget the procedures. Using the FOIL method Find ech product. ) (x 3 3)(x 3 6) ) (2 2 1)( 2 5) ) (x 3 3)(x 3 6) x 6 6x 3 3x 3 18 x 6 3x 3 18 ) (2 2 1)( 2 5) Multiplying Binomils Quickly The outer nd inner products in the FOIL method re often like terms, nd we cn comine them without writing them down. Once you ecome proficient t using FOIL, you cn find the product of two inomils without writing nything except the nswer. E X A M P L E 3 Using FOIL to find product quickly Find ech product. Write down only the nswer. ) (x 3)(x 4) ) (2x 1)(x 5) c) ( 6)( 6) ) (x 3)(x 4) x 2 7x 12 Comine like terms: 3x 4x 7x. ) (2x 1)(x 5) 2x 2 9x 5 Comine like terms: 10x x 9x. c) ( 6)( 6) 2 36 Comine like terms: E X A M P L E 4 More products Find ech product. ) 1 2 x x 1 ) (x 1)(x 3)(x 4) ) 1 2 x x x2 2 3 x 1 2 x x2 1 6 x ) Multiply the first two inomils nd then multiply tht result y x 4: (x 1)(x 3)(x 4) (x 2 2x 3)(x 4) x(x 2 2x 3) 4(x 2 2x 3) x 3 2x 2 3x 4x 2 8x 12 x 3 2x 2 11x 12
3 240 (5 18) Chpter 5 Polynomils nd Exponents E X A M P L E 5 Are of grden Sheil hs squre grden with sides of length x feet. If she increses the length y 7 feet nd decreses the width y 2 feet, then wht trinomil represents the re of the new rectngulr grden? The length of the new grden is x 7 nd the width is x 2 s shown in Fig The re is (x 7)(x 2) or x 2 5x 14 squre feet. x x 7 x x 2 FIGURE 5.2 WARM-UPS True or flse? Answer true only if the eqution is true for ll vlues of the vrile or vriles. Explin your nswer. 1. (x 3)(x 2) x (x 2)(y 1) xy x 2y 2 3. (3 5)(2 1) (y 3)(y 2) y 2 y 6 5. (x 2 2)(x 2 3) x 4 5x (3 2 2)(3 2 2) (t 3)(t 5) t 2 8t (y 9)(y 2) y 2 11y (x 4)(x 7) x 2 4x It is not necessry to lern FOIL s long s you cn get the nswer. 5.3 EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use complete sentences. 1. Wht property of the rel numers do we usully use to find the product of two inomils? 2. Wht does FOIL stnd for? 3. Wht is the purpose of FOIL? 4. Wht is the mximum numer of terms tht cn e otined when two inomils re multiplied? Use FOIL to find ech product. See Exmple (x 2)(x 4) 6. (x 3)(x 5) 7. ( 3)( 2) 8. ( 1)( 2) 9. (2x 1)(x 2)
4 5.3 Multipliction of Binomils (5 19) (2y 5)(y 2) 11. (2 3)( 1) 12. (3x 5)(x 4) 13. (w 50)(w 10) 14. (w 30)(w 20) 15. (y )(y 5) 16. ( t)(3 y) 17. (5 w)(w m) 18. ( h)( t) 19. (2m 3t)(5m 3t) 20. (2x 5y)(x y) 21. (5 2)(9 7) 22. (11x 3y)(x 4y) Use FOIL to find ech product. See Exmple (x 2 5)(x 2 2) 24. (y 2 1)(y 2 2) 25. (h 3 5)(h 3 5) 26. (y 6 1)(y 6 4) 27. (3 3 2)( 3 4) 28. (5n 4 1)(n 4 3) 29. (y 2 3)(y 2) 30. (x 1)(x 2 1) 31. (3m 3 n 2 )(2m 3 3n 2 ) 32. (6y 4 2z 2 )(6y 4 3z 2 ) 33. (3u 2 v 2)(4u 2 v 6) 34. (5y 3 w 2 z)(2y 3 w 2 3z) Find ech product. Try to write only the nswer. See Exmple ( 4)( 5) 36. (y 8)(y 4) 37. (x 3)(x 9) 38. (m 7)(m 8) 39. ( 5)( 5) 40. (t 4)(t 4) 41. (2x 1)(2x 1) 42. (3y 4)(3y 4) 43. (z 10)(z 10) 44. (3h 5)(3h 5) 45. ( )( ) 46. (x y)(x y) 47. ( )( 2) 48. ( 8c)( c) 49. (2x y)(x 3y) 50. (3y 5z)(y 3z) 51. (5t 2)(t 1) 52. (2t 3)(2t 1) 53. (h 7)(h 9) 54. (h 7w)(h 7w) 55. (h 7w)(h 7w) 56. (h 7q)(h 7q) 57. (2h 2 1)(2h 2 1) 58. (3h 2 1)(3h 2 1) Perform the indicted opertions. See Exmple x x t t x 4 (3x 1)(2x 5) 64. 4xy 3 (2x y)(3x y) 65. (x 1)(x 1)(x 3) 66. ( 3)( 4)( 5) 67. (3x 2)(3x 2)(x 5) 68. (x 6)(9x 4)(9x 4) 69. (x 1)(x 2) (x 3)(x 4) 70. (k 4)(k 9) (k 3)(k 7) Solve ech prolem. See Exmple Are of rug. Find trinomil tht represents the re of rectngulr rug whose sides re x 3 feet nd 2x 1 feet. 2x 1 FIGURE FOR EXERCISE 71 x Are of prllelogrm. Find trinomil tht represents the re of prllelogrm whose se is 3x 2 meters nd whose height is 2x 3 meters. 73. Are of sil. The sil of tll ship is tringulr in shpe with se of 4.57x 3 meters nd height of 2.3x 1.33 meters. Find polynomil tht represents the re of the tringle. 74. Are of squre. A squre hs side of length 1.732x meters. Find polynomil tht represents its re.
5 242 (5 20) Chpter 5 Polynomils nd Exponents GETTING MORE INVOLVED 75. Explortion. Find the re of ech of the four regions shown in the figure. Wht is the totl re of the four regions? Wht does this exercise illustrte? 76. Explortion. Find the re of ech of the four regions shown in the figure. Wht is the totl re of the four regions? Wht does this exercise illustrte? h ft 4 ft h ft h ft 3 ft 3 ft 4 ft h ft FIGURE FOR EXERCISE 75 FIGURE FOR EXERCISE 76 In this section The Squre of Binomil Product of Sum nd Difference Higher Powers of Binomils Applictions to Are 5.4 SPECIAL PRODUCTS In Section 5.3 you lerned the FOIL method to mke multiplying inomils simpler. In this section you will lern rules for squring inomils nd for finding the product of sum nd difference. These products re clled specil products. The Squre of Binomil To compute ( ) 2, the squre of inomil, we cn write it s ( )( ) nd use FOIL: ( ) 2 ( )( ) So to squre, we squre the first term ( 2 ), dd twice the product of the two terms (2), then dd the squre of the lst term ( 2 ). The squre of inomil occurs so frequently tht it is helpful to lern this new rule to find it. The rule for squring sum is given symoliclly s follows. The Squre of Sum ( ) E X A M P L E 1 Using the rule for squring sum Find the squre of ech sum. ) (x 3) 2 ) (2 5) 2
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