MULTIPLICATION OF BINOMIALS 4.3. section. The FOIL Method

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1 4. Multipliction of Binomils (4-5) 98. Ptchwork. A quilt ptch cut in the shpe of tringle hs se of 5x inches nd height of.7x inches. Wht polynomil represents its re? 4.x squre inches FIGURE FOR EXERCISE Totl revenue. If promoter chrges p dollrs per ticket for concert in Tuls, then she expects to sell 40, p tickets to the concert. How mny tickets will she sell if the tickets re $0 ech? Find the totl revenue when the tickets re $0 ech. Wht polynomil represents the totl revenue expected for the concert when the tickets re p dollrs ech? 0,000, $00,000, 40,000p 000p 00. Mnufcturing shirts. If mnufcturer chrges p dollrs ech for rugy shirts, then he expects to sell p shirts per week. Wht polynomil represents the totl revenue expected for week? How mny shirts will e sold if the mnufcturer chrges $0 ech for the shirts? Find the totl revenue when the shirts re sold for $0 ech. Use the r grph to determine the price tht will give the mximum totl revenue. 000p 00p, 0, $0, $0 5x.7x Totl revenue (thousnds of dollrs) Price (dollrs) FIGURE FOR EXERCISE Periodic deposits. At the eginning of ech yer for 5 yers, n investor invests $0 in mutul fund with n verge nnul return of r. If we let x r, then t the end of the first yer (just efore the next investment) the vlue is 0x dollrs. Becuse $0 is then dded to the 0x dollrs, the mount t the end of the second yer is (0x 0)x dollrs. Find polynomil tht represents the vlue of the investment t the end of the fifth yer. Evlute this polynomil if r 0%. 0x 5 0x 4 0x 0x 0x, $ Incresing deposits. At the eginning of ech yer for 5 yers, n investor invests in mutul fund with n verge nnul return of r. The first yer, she invests $0; the second yer, she invests $0; the third yer, she invests $0; the fourth yer, she invests $40; the fifth yer, she invests $50. Let x r s in Exercise 0 nd write polynomil in x tht represents the vlue of the investment t the end of the fifth yer. Evlute this polynomil for r 8%. 0x 5 0x 4 0x 40x 50x, $80.5 GETTING MORE INVOLVED 0. Discussion. Nme ll properties of the rel numers tht re used in finding the following products: ) c 5 c ) (x )(x 8x 6) 04. Discussion. Find the product of 7 nd 46 without using clcultor. Then use the distriutive property to find the product (0 7)( ) s you would find the product of inomil nd trinomil. Explin how the two methods re relted. 4. MULTIPLICATION OF BINOMIALS In this section The FOIL Method Multiplying Binomils Quickly In Section 4. 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 )(x ) (x )x (x ) Distriutive property x x x 6 Distriutive property x 5x 6 Comine like terms.

2 (4-6) Chpter 4 Polynomils nd Exponents There re four terms in x x x 6. The term x is the product of the first term of ech inomil, x nd x. The term x is the product of the two outer terms, nd x. The term x is the product of the two inner terms, nd x. The term 6 is the product of the lst term of ech inomil, nd. We cn connect the terms multiplied y lines s follows: F (x )(x ) I O L 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 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 5. Using the FOIL method Find ech product. ) (x )(x 4) ) (x 5)(x 4) c) ( )( ) d) (x )(y 5) F ) (x )(x 4) x 4x x 8 x x 8 I O L F O I L Comine the like terms. ) (x 5)(x 4) 6x 8x 5x 0 6x 7x 0 Comine the like terms. c) ( )( ) d) (x )(y 5) xy 5x y 5 There re no like terms to comine. FOIL cn e used to multiply ny two inomils. The inomils in the next exmple hve higher powers thn those of Exmple. E X A M P L E 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 )(x 6) ) ( )( 5) ) (x )(x 6) x 6 6x x 8 x 6 x 8 ) ( )( 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.

3 4. Multipliction of Binomils (4-7) E X A M P L E Using FOIL to find product quickly Find ech product. Write down only the nswer. ) (x )(x 4) ) (x )(x 5) c) ( 6)( 6) ) (x )(x 4) x 7x Comine like terms: x 4x 7x. ) (x )(x 5) x 9x 5 Comine like terms: 0x x 9x. c) ( 6)( 6) 6 Comine like terms: x E X A M P L E 4 x x 7 x FIGURE 4. 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 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 s shown in Fig. 4.. The re is (x 7)(x ) or x 5x 4 squre feet. WARM-UPS True or flse? Answer true only if the eqution is true for ll vlues of the vrile or vriles. Explin your nswer.. (x )(x ) x 6 Flse. (x )(y ) xy x y True. ( 5)( ) True 4. (y )(y ) y y 6 True 5. (x )(x ) x 4 5x 6 True 6. ( )( ) 9 4 Flse 7. (t )(t 5) t 8t 5 True 8. (y 9)(y ) y y 8 Flse 9. (x 4)(x 7) x 4x 8 Flse 0. It is not necessry to lern FOIL s long s you cn get the nswer. Flse 4. EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use complete sentences.. Wht property of the rel numers do we usully use to find the product of two inomils? We use the distriutive property to find the product of two inomils.. Wht does FOIL stnd for? FOIL stnds for first, outer, inner, nd lst.. Wht is the purpose of FOIL? The purpose of FOIL is to provide fster method for finding the product of two inomils. 4. Wht is the mximum numer of terms tht cn e otined when two inomils re multiplied? The mximum numer of terms otined in multiplying inomils is four. Use FOIL to find ech product. See Exmple. 5. (x )(x 4) x 6x 8 6. (x )(x 5) x 8x 5 7. ( )( ) 6 8. ( )( ) 9. (x )(x ) x 5x

4 4 (4-8) Chpter 4 Polynomils nd Exponents 0. (y 5)(y ) y 9y 0. ( )( ). (x 5)(x 4) x 7x 0. (w 50)(w 0) w 60w (w 0)(w 0) w 50w (y )(y 5) y y 5y 5 6. ( t)( y) t y ty 7. (5 w)(w m) 5w w 5m mw 8. ( h)( t) h t ht 9. (m t)(5m t) 0m 9mt 9t 0. (x 5y)(x y) x xy 5y. (5 )(9 7) (x y)(x 4y) x 47xy y Use FOIL to find ech product. See Exmple.. (x 5)(x ) x 4 x 0 4. (y )(y ) y 4 y 5. (h 5)(h 5) h 6 0h 5 6. (y 6 )(y 6 4) y y ( )( 4) (5n 4 )(n 4 ) 5n 8 4n 4 9. (y )(y ) y y y 6 0. (x )(x ) x x x. (m n )(m n ) 6m 6 7m n n 4. (6y 4 z )(6y 4 z ) 6y 8 0y 4 z 6z 4. (u v )(4u v 6) u 4 v 0u v 4. (5y w z)(y w z) 0y 6 w 4 7y w z z Find ech product. Try to write only the nswer. See Exmple. 5. ( 4)( 5) (y 8)(y 4) y y 7. (x )(x 9) x 6x 7 8. (m 7)(m 8) m m ( 5)( 5) (t 4)(t 4) t 8t 6 4. (x )(x ) 4x 4x 4. (y 4)(y 4) 9y 4y 6 4. (z 0)(z 0) z (h 5)(h 5) 9h ( )( ) 46. (x y)(x y) x xy y 47. ( )( ) 48. ( 8)( ) (x )(x ) x 5x 50. (y 5)(y ) y 4y 5 5. (5t )(t ) 5t 7t 5. (t )(t ) 4t 8t 5. (h 7)(h 9) h 6h (h 7w)(h 7w) h 4hw 49w 55. (h 7w)(h 7w) h 4hw 49w 56. (h 7q)(h 7q) h 49q 57. (h )(h ) 4h 4 4h 58. (h )(h ) 9h 4 6h Perform the indicted opertions x x 8 x 6 x t t t 4 t x 4 (x )(x 5) x 6 6x 5 0x xy (x y)(x y) 4x y 4x y 4 4xy (x )(x )(x ) x x x 66. ( )( 4)( 5) (x )(x )(x 5) 9x 45x 4x (x 6)(9x 4)(9x 4) 8x 486x 6x (x )(x ) (x )(x 4) x (k 4)(k 9) (k )(k 7) k 5 Solve ech prolem. 7. Are of rug. Find trinomil tht represents the re of rectngulr rug whose sides re x feet nd x feet. x 5x squre feet x FIGURE FOR EXERCISE 7 x + 7. Are of prllelogrm. Find trinomil tht represents the re of prllelogrm whose se is x meters nd whose height is x meters. 6x x 6 squre meters 7. Are of sil. The sil of tll ship is tringulr in shpe with se of 4.57x meters nd height of.x. meters. Find polynomil tht represents the re of the tringle x x.995 squre meters 74. Are of squre. A squre hs side of length.7x.44 meters. Find polynomil tht represents its re..9998x 4.898x.999 squre meters

5 4.4 Specil Products (4-9) 5 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? ft, h ft, 4h ft, h ft, h 7h ft, (h )(h 4) h 7h 4 ft h ft 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 h ft ft 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 helpful hint To visulize the squre of sum, drw squre with sides of length s shown. 4.4 SPECIAL PRODUCTS In Section 4. 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 ( ), the squre of inomil, we cn write it s ( )( ) nd use FOIL: ( ) ( )( ) So to squre, we squre the first term ( ), dd twice the product of the two terms (), then dd the squre of the lst term ( ). 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 re of the lrge squre is ( ). It comes from four terms s stted in the rule for the squre of sum. The Squre of Sum ( ) E X A M P L E Using the rule for squring sum Find the squre of ech sum. ) (x ) ) ( 5)