7 ()7 () 7 (7.5 )1. (a) 3 , 2 (b) 3 , 4 (c) 4 , 3 (d) 4 , 7 (e) 5 , 3 (f) 1 , 0 2. (a)1 5 23 4 5) 3 ( ) 4 5 (2 32 3 32 3 3+ + + + + + + + x x x x x x x x x x x(b)1 3 2 42 1 3 3 2 6) 2 3 2 ( ) 1 3 6 (2 32 2 3 32 3 2 3 + + + + + + + + +x x x x x x x x x x x x x(c)5 4 61 4 6 2 3 3) 1 6 3 ( ) 4 3 2 (22 22 2+ + + + + + x x x x x x x x x x(d)7 13 23 4 6 7 4 3 2) 3 6 4 ( ) 4 2 7 3 (2 32 2 32 3 2+ + + + + + + + +x x x x x x x x x x x x x3. (a) ) 4 )( 3 (2x x x + +x x x x x x x x x x x x12 712 3 4) 4 ( 3 ) 4 (2 32 2 32 2+ + + + + + + + (b) ) 1 )( 1 3 4 (2 + + x x x1 2 41 3 4 3 4) 1 )( 1 3 4 ( ) 1 3 4 (2 32 2 32 3 + + + + + + + x x x x x x x x x x x x x(c) ) 5 2 )( 3 2 (2 x x x10 17 7 615 6 3 10 4 2) 5 2 ( 3 ) 5 2 ( 22 33 2 22 2 + + + x x x x x x x x x x x x x4. (a) ) 1 2 2 5 ( ) 3 4 5 ( ) 7 3 4 (2 3 3 2 3 + + + x x x x x x x3 22 3 3 2 35 7 51 2 2 5 3 4 5 7 3 4x x x x x x x x x x+ + + + (b)) 10 2 3 4 () 1 2 ( ) 4 6 5 (3 23 2 3 + + + + x x x x x x x x3 23 2 3 2 310 1310 2 3 4 1 2 4 6 5x x x x x x x x x x x+ + + + 5. (a) ) 1 3 )( 1 3 (2 2+ + + x x x x4 3 22 3 42 2 42 2 26 9 11 9 61 ) 3 ( ) 3 ( 21 ) 3 (x x x x x x x x x x x x+ + + + + + + + (b) ) 1 3 5 )( 3 2 1 (2 2 + + x x x x4 3 24 3 23 2 22 22 22 26 19 16 2 16 9 310 15 5 2 3 1) 3 )( 2 3 1 ( ) 5 )( 2 3 1 ( ) 1 )( 2 3 1 () 3 5 1 )( 2 3 1 (x x x xx x xx x x x xx x xx x x x xx x x x+ + + + + + + + + + + + + 6. (a)2 2) 3 4 ( ) 4 3 ( + x x227 48 77 49 7) 7 )( 1 7 ()] 3 4 ( ) 4 3 )][( 3 4 ( ) 4 3 [(x x x x x x x x x x x + + + + + + + + (b)2 2) 1 2 ( 2 ) 1 )( 1 2 ( + + + + x x x x3 22 2 32 22222 5 5 11 3 2 6 2) 1 3 ( ) 1 3 ( 2) 1 3 )( 1 2 () 2 4 1 )( 1 2 ()] 1 2 ( 2 ) 1 )[( 1 2 (x x x x x x x x x x x x x x x x x x x x x x x x+ + + + + + + + + + + 7. (a) 11 4 12 40 321 ) 2 ( 2 ) 2 ( 3 ) 2 ( 5 ) 2 ( 22 3 4+ + + + + + (b) 231 143858112122132152122 3 4 + + ,_

+ ,_

,_

+ ,_

8. (a)2118 2 518 ) 1 ( 2 ) 1 ( 5 ) 1 (3+ + f(b)11118 6 13518 ) 3 ( 2 ) 3 ( 5 ) 3 (3 + + + f1 4B 9.a a a h+ + + + + 2112 9) 3 ( 4 ) 3 ( ) 3 (2 h(3) = 14714 21 +aa 7.1 (7.27 )1.ac x c ab x b a x ac cx abx bx ax x c bx x a x x f+ + + + + + + + + + + + + ) ( ) () )( ( ) (2 32 2 32ac = 6 a c 62. a 123 6a f(x) 3. x 1 x + 1 x 2 x + 2 x 3 x + 3 x 4 x + 4 x 5 x + 5 x 6 x + 64. 06 5 2 16 ) 1 ( 5 ) 1 ( 2 1 ) 1 (2 3+ + f x 1 f(x) 086 5 2 16 ) 1 ( 5 ) 1 ( 2 ) 1 ( ) 1 (2 3+ + + f x + 1 f(x) 046 10 8 86 ) 2 ( 5 ) 2 ( 2 2 ) 2 (2 3 + + f x 2 f(x) 06 10 8 86 ) 2 ( 5 ) 2 ( 2 ) 2 ( ) 2 (2 3+ + + f x + 2 f(x) 06 15 18 276 ) 3 ( 5 ) 3 ( 2 3 ) 3 (2 3+ + f x 3 f(x) 0246 15 18 276 ) 3 ( 5 ) 3 ( 2 ) 3 ( ) 3 (2 3 + + + f x + 3 f(x) 01206 30 72 2166 ) 6 ( 5 ) 6 ( 2 6 ) 6 (2 3+ + f x 6 f(x) 02526 30 72 2166 ) 6 ( 5 ) 6 ( 2 ) 6 ( ) 6 (2 3 + + + f x + 6 f(x) f(x) x 3x 1 x + 2()5. f(x) x 3x 1 x + 2) 2 )( 1 )( 3 ( ) ( + x x x x f (7.31 ) (7.7 )1. (a)xxxx x xx2 2 32 32 3 22(b)1 6 6 8 8 61 6 8 6 23 4 3 2232 32 + ++ +xxxxxx x x xx x2. (a) 3 3 53 53 5 2232 32xxxxxx x x xx x+ + = 3 52+ x x =027 ()(b)1 4 4 4 4 21 4 4 2 22 2 2232 32xxxxxx x x xx x+ + + = 2 22 + x x =1(c)1 4 4 61 4 6 22 3 2242 4 22++ + xxxxx x x xx = 2 32 x =1 + x (7.14 )(a) ) 1 ( f (b) ) 3 ( f(c) ) 5 ( f (d) ) 6 ( f (7.17 )(a) ,_

23f(b) ,_

32f(c) ) 0 ( f (d) ) 3 ( f 7.1 (7.8 )10 31520 105 10 6 35 4 3 222+ + xxxx xx x x =10 3 + x =15 7.2 (7.9 )2 4 45 2 27 2 4 4 6 4 4 47 6 0 4 12222 32 3 ++ + + + x xxxx xx xx xx x x x = 2 4 42 + x x =5 7.3 (7.9 )3 56 123 3 33 9 3 5 5 53 4 2 5 1222 32 3 2+ + + + + +xxx xx xx x xx x x x x =3 5 x =6 12 x 7.4 (7.11 )f(x)7 13 48 1 3 4 12 48 ) 1 )( 1 3 ( ) 4 )( 1 3 (8 ) 1 4 )( 1 3 ( ) (2 32 2 32 22+ + + + + + + + + + + + + + + + x x x x x x x x x x x x x x x x x f 4x3 + 13x2 + x + 7 7.5 (7.12 )p(x)42) 2 )( 4 (28 2211 ) 3 2 () (11 ) 2 ( ) ( 3 2222+ + + + ++ + +x x x x x x x xx xx px x p x x x + 43 4B 7.6 (7.15 ) f(x) = x3 3x2 + 4x 2 (a) 02 ) 1 ( 4 ) 1 ( 3 ) 1 () 1 (2 3 + f(b) 302 ) 2 ( 4 ) 2 ( 3 ) 2 () 2 (2 3 + f 7.7 (7.15 ) f(x) = x3 2x2 + qx + 724 64 7 ) 1 ( ) 1 ( 2 ) 1 (4 ) 1 (2 3 + + + qqq f 7.8 (7.16 ) f(x) = 2x2 + 3x + 939 32 9 ) ( 3 ) ( 22 ) (2 22 + + bb b b b b b f 7.9 (7.18 ) f(x) = 4x2 6x + 3(a) 213236234232+ ,_

,_

,_

f(b) 25313526524522+ ,_

,_

,_

f 7.10 (7.18 ) f(x) = 16x3 + 8x2 + cx 516827 52121821167212 3 ,_

+ ,_

+ ,_

,_

cccf 7.11 (7.19 ) f(x) = x3 + cx2 + 3x + d f(x) x 2 ) 1 ( 28 414 ) 2 ( 3 ) 2 ( ) 2 (14 ) 2 (2 3 + + + + d c d c f f(x) x + 1 ) 2 ( 1317 ) 1 ( 3 ) 1 ( ) 1 (17 ) 1 (2 3 + + + + d c d c f(1) (2) 515 3 cc c = 5 (2)813 5 + dd 7.12 (7.22 ) f(x) = 8x3 4x2 + 2x 1(a)0851 ) 2 ( 2 ) 2 ( 4 ) 2 ( 8 ) 2 (2 3 + f x + 2 8x3 4x2 + 2x 1 (b)01 1 1 11212214218212 3 + ,_

+ ,_

,_

,_

f 2x 1 8x3 4x2 + 2x 1 7.13 (7.23 ) f(x) = 4x3 + kx2 17x 3 f(x) 4x 3 25162251690 34317434340432 3 ,_

,_

+ ,_

,_

kkkf 7.14 (7.24 )(a) 012 24 9 2712 ) 3 ( 8 ) 3 ( ) 3 ( ) 3 (2 3+ + + f x + 3 f(x) 47 ()(b) 12 4 12 4 12 4 8 4 312 8 34 4 222 32 32++ ++ ++ xxx xx xx xx x x xx xf(x) = (x + 3)(x2 4x + 4)x2 4x + 4 = (x 2)2 f(x) =2) 2 )( 3 ( + x x 7.15 (7.24 ) f(x) = ax3 + 2bx2 29x 6 Q(x)) ( ) 2 )( 3 () ( ) 6 ( ) (2x Q x x x Q x x x f + + f(x) x + 3 x 2 f(3) = 0 f(2) = 0) 1 ( 9 2 30 81 18 270 6 ) 3 ( 29 ) 3 ( 2 ) 3 (0 ) 3 (2 3 + + + + b ab ab a f) 2 ( 80 64 8 80 6 ) 2 ( 29 ) 2 ( 2 ) 2 (0 ) 2 (2 3 + + +b ab ab a f(2) 2 (1)525 5aaa = 5 (2)38 5 +bb 7.16 (7.29 )0 12 ) 1 ( 5 ) 1 ( 6 ) 1 ( ) 1 (0 12 12 ) 1 ( 5 ) 1 ( 6 ) 1 ( ) 1 (2 32 3 + + + + f f x + 1 f(x) 12 12 12 12 7 7 5 7 12 5 6 112 7 222 32 32++ + ++ + ++ xxx xx xx xx x x xx x) 4 )( 3 )( 1 () 12 7 )( 1 ( ) (2 + + + x x x x x x x f 7.17 (7.29 )3x3 3x2 + 6x + 12 = 3(x3 x2 + 2x + 4) g(x) = x3 x2 + 2x + 40 4 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 (0 6 4 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 (2 32 3 + + + + gg x + 1 g(x) ) 4 2 )( 1 ( 3 12 6 3 3) 4 2 )( 1 ( 4 22 2 32 2 3+ + + + + + + + x x x x x x x x x x x x 7.18 (7.30 ) f(x) = 2x3 17x2 + 38x 15015 19417411521382117212212 3 + ,_

+ ,_

,_

,_

f 2x 1 f(x) ) 5 )( 3 )( 1 2 () 15 8 )( 1 2 ( 15 38 17 22 2 3 + + x x x x x x x x x (7.10 )1.7 2 4 5 4 3 65 7 6 1 22 3 22++ + ++xxx xx x xx =2 3 + x =7 2.68 69 23 1 23 21 7 2 7 31 2 4 323 7 222 32 32++ + ++ + ++ xxx xx xx xx x x xx x = 23 72+ x x =68 5 4B 3. 5 212 35 5 107 2 10 2 2 47 0 8 4 1 2222 32 3 2+ + + ++ + + +xxx xx xx x xx x x x x =5 2 x =12 3 + x (7.16 )1. f(x) = 2x3 + mx2 mx 4318 632 4 ) 3 ( ) 3 ( ) 3 ( 232 ) 3 (2 3 +mmm m f2. f(x) = x2 + 2x 828 44 8 ) 2 ( 2 ) 2 (4 ) 2 (2 22 +nn n n n n n f (7.19 )1. (a) f(x) = 27x3 18x2 + 9x 5 275 6 8 8532932183227322 3 ,_

+ ,_

,_

,_

f(b) f(x) = x4 + 2x2 x + 116171212116112121221212 4+ + + ,_

,_

+ ,_

,_

f2. f(x) = 2x2 + (k 1)x + 9189292 2 292) 1 (2222 22 + + + ,_

+ ,_

,_

kkkk k kkkkkkkf3. f(x) = x3 + 2x2 + ax + b f(x) x 1 ) 1 ( 12 ) 1 ( ) 1 ( 2 ) 1 (2 ) 1 (2 3 + + + +b a b a f f(x) x + 2 ) 2 ( 10 210 ) 2 ( ) 2 ( 2 ) 2 (10 ) 2 (2 3 + + + + b a b a f(1) (2) 39 3aa a = 3 (1)41 3 +bb4. f(x) = 3x3 + 2x2 + cx + d f(x) 2x 1 ) 1 ( 32181721212213817212 3 + + ,_

+ ,_

+ ,_

,_

d cd cf f(x) x + 2 ) 2 ( 28 244 ) 2 ( ) 2 ( 2 ) 2 ( 344 ) 2 (2 3 + + + + d c d c f(1) (2) 102525cc c = 10 (1)83 ) 10 (21 +dd (7.25 )1. f(x) = x3 8x2 + 17x 10(a)010 34 32 810 ) 2 ( 17 ) 2 ( 8 ) 2 ( ) 2 (2 3 + + f x 2 x3 8x2 + 17x 10 67 ()(b)042010 85 200 12510 ) 5 ( 17 ) 5 ( 8 ) 5 ( ) 5 (2 3 + f x + 5 x3 8x2 + 17x 102. f(x) = 2x3 5x2 x + 1(a)0211214541121215212212 3 + + ,_

,_

,_

,_

f 2x 1 2x3 5x2 x + 1 (b)0231123445427123235232232 3 + + + ,_

,_

,_

,_

f 2x + 3 2x3 5x2 x + 13. f(x) = 2kx3 + x2 4x + 1 f(x) 2x + 1 1341340 1214212120212 3 + ,_

,_

+ ,_

,_

kkkf4. (a) f(3) = 2(3)3 8(3)2 + 7(3) 3 = 54 72 + 21 3 = 0 x 3 f(x) (b) 3 3 6 2 7 2 6 23 7 8 2 31 2 2 222 32 32+ + + + xxx xx xx xx x x xx x f(x) =) 1 2 2 )( 3 (2+ x x x (7.30 )1. f(x) = x3 + x2 34x + 56 f(1) = (1)3 + (1)2 34(1) + 56 = 24f(1) = (1)3 + (1)2 34(1) + 56 = 90f(2) = (2)3 + (2)2 34(2) + 56 = 0 x 2 f(x) ) 7 )( 4 )( 2 () 28 3 )( 2 ( 56 342 2 3+ + + +x x x x x x x x x2. f(x) = x3 + x2 21x 45 f(1) = (1)3 + (1)2 21(1) 45 = 64f(1)= (1)3 + (1)2 21(1) 45 = 24f(3) = (3)3 + (3)2 21(3) 45 = 72f(3)= (3)3 + (3)2 21(3) 45 = 0 x + 3 f(x) 22 2 3) 3 )( 5 () 3 )( 5 )( 3 () 15 2 )( 3 ( 45 21+ + + + +x x x x x x x x x x x3. 2x3 + 2x2 16x 24 = 2(x3 + x2 8x 12) f(x) = x3 + x2 8x 12 f(1) = (1)3 + (1)2 8(1) 12 = 18f(1)= (1)3 + (1)2 8(1) 12 = 4f(2) = (2)3 + (2)2 8(2) 12 = 16f(2)= (2)3 + (2)2 8(2) 12 = 0 x + 2 f(x) 2 2 322 2 3) 2 )( 3 ( 2 24 16 2 2) 2 )( 3 () 2 )( 3 )( 2 () 6 )( 2 ( 12 8+ ++ + + + +x x x x x x x x x x x x x x x x4. f(x) = 3x3 + 13x2 51x + 35 f(1) = 3(1)3 + 13(1)2 51(1) + 35 = 0 x 1 f(x) ) 7 )( 5 3 )( 1 () 35 16 3 )( 1 ( 35 51 13 32 2 3+ + + +x x x x x x x x x 7A (7.12 )1.3 4 4 23 4 2 2222xxxx x xx+ =2 x =37 4B 2.3 2 2 5 2 5 3 1222++++ + ++xxx xx x xx =2 + x =33.1 4 21 4 2 22 22x xx x xx+ = x 2 =14.8 6 28 6 2 32 22x x x x xx+ = x 2 =85.1 2 1 2 2 41 4 4 1 21 2 22+ + + xxx xx x xx =1 2 x =06.3 1 4 4 4 4 164 0 16 1 41 4 22 + + +xxx xx x xx =1 4 x =3 7.4 20 12 16 12 5 316 7 3 5 34 22 + +xxx xx x xx =4 x =48.10 15 5 5 5 9 35 4 3 35 3 22+ + + + + + +xxx xx x xx =5 3 + x =10 9.25 30 10 5 10 9 3 3 6 25 3 2 310 3 2 222 32 32+ + + +xxx xx xx xx x x xx x 2x3 3x2 + x 5 = (x 3)(2x2 + 3x + 10) + 2510.7 2 3 5 3 4 6 7 6 6 95 7 12 9 2 31 2 3 222 32 32++++ + + ++ +xxx x x xx x x x x xx x 9x3 + 12x2 + 7x 5 = (3x + 2)(3x2 + 2x + 1) + (7)11.9 2 4 7 4 4 8 0 8 2 47 0 6 4 1 22 4 2 222 32 32+ + + + ++ xxx xx xx xx x x xx x 4x3 6x2 7 = (2x + 1)(2x2 4x + 2) + (9)87 ()12.1 12 6 11 6 24 12 30 12 12 611 30 0 6 12 61 2 222 32 32+ + + + + + xxx xx xx xx x x xx x 30x 6x3 11 = (12 + 6x)(x2 2x + 1) + 113. f(x)12 5 618 6 4 9 618 ) 2 )( 3 2 ( ) 3 )( 3 2 (18 ) 2 3 )( 3 2 ( ) (22+ + + + + + + + + + x x x x x x x x x x x f 6x2 + 5x + 1214. f(x)14 23 66 20 15 8 6) 6 ( ) 5 )( 4 3 ( ) 2 )( 4 3 () 6 ( ) 5 2 )( 4 3 ( ) (22+ + + + + x x x x x x x x x x x f 6x2 23x + 1415. f(x)3 27 23 67 10 12 15 18 5 67 ) 2 )( 5 6 ( ) 3 )( 5 6 ( ) )( 5 6 (7 ) 2 3 )( 5 6 ( ) (2 32 2 322 + + + + + + + + + x x x x x x x x x x x x x x x x x f 6x3 23x2 + 27x 316. p(x)2 32) 2 )( 2 3 (24 4 32) 3 ( ) 7 4 3 () () 3 ( ) 2 ( ) ( 7 4 3222 ++ + ++ + + + +x x x x x x x xx xx px x p x x 3x 217. p(x)5 21 2) 5 2 )( 1 2 (1 25 12 41 23 ) 2 12 4 () (3 ) 1 2 ( ) ( 2 12 4222+ + + + + + x x x x x x x x x xx px x p x x 2x + 518. ) 5 ( 3 52 5 5) 2 ( ) 5 )( 1 ( ) )( 1 () 2 ( ) 5 )( 1 ( 2 52 32 32 22 2 3 + + + + + + + + + + + + +b x x x b x x x x b x x x x b x x x ax x xx a =3 72 5 bb19. 6 ) 16 ( 5 318 12 2 2 18 3 3) 18 ( ) 2 )( 6 ( ) 3 )( 6 () 18 ( ) 2 3 )( 6 (5 32 32 2 32 222 3 + + + + + + + + + x q x x qx x x x x x qx x x x x x qx x x x p x x xp =6 x 171 16 qq20.2 20 15 5 18 16 5 8 6 218 24 2 4 35 2 222 32 3 2+ +xx xx xx x xx x x x xx =5 2 + x =2 + x9 4B 21.1 3 5 10 15 4 13 15 3 6 94 16 9 9 1 2 35 3 222 32 3 2+ + +xx xx xx x xx x x x xx =5 3 + x =1 3 + x22.5 2 6 3 12 1 5 12 4 2 81 9 10 8 2 43 2 222 32 3 2 + + ++ +xx xx xx x xx x x x xx =3 2 x =5 2 x23.2 1 1 1 0 0 11 222 32 3 2 + + ++ + + + +x xx xx x xx x x x xx =1 x =224.15 68 20 60 20 5 8 20 5 15 5 3 5 5 35 3 4 2 1 320 5 222 32 32 3 42 3 4 22+ + + + + + ++ xx xx xx x xx x xx x xx x x x x xx x =20 52+ x x =15 68 + x25. (3x3 + 2x 3) (x3 4x2 + 4) = 2x3 + 4x2 + 2x 757 22 50 30 10 7 8 0 1 10 6 27 2 4 2 5 310 2 222 32 3 2+ + + + + +xx xx xx x xx x x x xx =10 2 + x =57 22 x26. (a) 10 16 28 5 618 3 8 3 216 6 4 24 9 6) 18 3 ( ) 1 )( 8 3 2 () 2 )( 8 3 2 ( ) 3 )( 8 3 2 () 18 3 ( ) 1 2 3 )( 8 3 2 ( ) (2 3 422 3 2 3 422 2 22 2+ + + + ++ + + + + ++ + + + + + + + x x x x x x x x x x x x x x x x x x x x x x x x x x x x f 10 16 28 5 6 ) (2 3 4+ + + x x x x x f(b)12 2 4 10 4 12 24 16 24 4 8 28 8 3 610 16 28 5 6 1 22 12 4 3 222 32 33 42 3 42 3++ + + + + + + xxx x x xx x x xx x x x x x xx x x = 2 12 4 32 3 + x x x =1227. p(x)618 9 46) 17 10 ( ) 1 4 () () 17 10 ( ) 6 ( ) ( 1 42 32 32 3 + + + + + x x x x x x x x xx px x x p x x x18 3 18 3 12 2 9 2 618 9 4 63 2 222 32 32 + +xxx xx xx xx x x xx x x2 + 2x + 3107 ()28. p(x)1 3 22 9 13 61 3 216 ) 18 9 13 6 () (16 ) 1 3 2 ( ) ( 18 9 13 622 322 32 2 3+ + + + + + + + + + x x x x x x x x x xx px x x p x x x 2 6 4 2 6 4 3 9 62 9 13 6 1 3 22 3 222 32 3 2+ + + + + + + x xx xx x xx x x x xx 3x + 229. (a) 36 113) 1 13 ( ) 5 2 () () 1 13 ( ) 3 ( ) ( 5 2333+ ++ ++ + + +x x x x x x xx px x x p x x 6 2 6 2 9 3 11 3 36 11 0 32 3 222 32 32 + + + xxx xx xx xx x x xx x 2 3 ) (2 x x x p(b)3 3 10 15 5 7 12 5 12 18 6 0 23 6 4 6 27 0 19 12 2 2 35 6 2 222 32 32 3 42 3 4 22 + + + + + + + + + + + xx x x x x x x x x xx x x x x x x x xx x = 5 6 22 + x x =3 3 x30. ) 6 ( ) 8 3 ( ) 3 2 ( 212 6 3 3 4 2 2) 12 ( ) 3 )( 2 ( ) 2 )( 2 () 12 ( ) 3 2 )( 2 (4 3 22 32 2 32 222 3 + + + + + + + + + + + + + + + + + + + + + c x b x b x c x bx x x bx x c x bx x x bx x c x x bx x ax x xx233 2 3 + b bx a = 3(3) + 8 =1 104 6 cc31. ) ( ) 1 ( ) 1 2 ( 22 2) ( ) )( 1 2 ( ) )( 1 2 () ( ) )( 1 2 (3 5 42 32 2 32 222 3d b x c b x b x d cx b bx bx x x x d cx b x x x x x d cx b x x x x x ax+ + + + + + + + + + + + + + + + + +x3a =2x2231 2 4+ bbx 291235 + cc23233

,_

+ dd32. Q(x)3x3 16x2 + ax + b = (x 1)(x 3) Q(x) + (x + 2) x = 1) 1 ......( 163 13) 2 1 ( ) 1 ( ) 3 1 )( 1 1 ( ) 1 ( ) 1 ( 16 ) 1 ( 32 3 + + + + + + + b a b a Q b a x = 3) 2 ......( 68 35 3 63) 2 3 ( ) 3 ( ) 3 3 )( 1 3 ( ) 3 ( ) 3 ( 16 ) 3 ( 32 3 + + + + + + + b a b a Q b a(2) (1)2652 216 68 ) ( ) 3 ( + +aab a b a a = 26 (1)1016 26 +bb11 4B 33. ' + ) 2 ( ) ( ) 4 5 ( ) ( ) () 1 ( ) 1 ( ) ( ) 4 5 ( ) (21 x Q x x g x f x Q x x f Q1(x) Q2(x) x (1) (2)[ ] [ ][ ]1 ) ( ) 4 5 (1 ) ( ) ( ) 4 5 ( ) () ( ) 4 5 () 1 ( ) ( ) 4 5 ( ) ( ) ( ) (2 121 + x Q x x Q x Q x x g x Q x x Q x x g x f x f Q(x) = Q1(x) Q2(x) Q(x) = 0 g(x) = 1 Q(x) = 1 g(x) = 1 (5x 4) Q(x) = 2 g(x) = 1 2(5x 4)() 7B (7.19 )1. f(x) = x3 8x 747 ) 3 ( 8 ) 3 () 3 (3 f2. f(x) = 2x3 x2 + 7x + 191 ) 1 ( 7 ) 1 ( ) 1 ( 2) 1 (2 3 + + f3. f(x) = x3 + 7x2 5x + 1271 ) 2 ( 5 ) 2 ( 7 ) 2 () 2 (2 3+ + f4. f(x) = 2x3 3x2 4x + 5645 ) 3 ( 4 ) 3 ( 3 ) 3 ( 2) 3 (2 3 + f5. f(x) = 8x3 2x + 111212218213+ ,_

,_

,_

f6. f(x) = 4x3 x2 + x 1316134343434432 3 ,_

+ ,_

,_

,_

f7. f(x) = 27x3 + 18x2 6x + 56531631183127312 3+ ,_

,_

+ ,_

,_

f8. f(x) = 2x3 + 2x2 4x + 74497234232232232 3+ ,_

,_

+ ,_

,_

f9. f(x) = 4x2(x 2) 10x + 721172510 225254252 + ,_

1]1

,_

,_

,_

f10. f(x) = (3x2 5)(1 x) 6x 9929346341 5343342 ,_

1]1

,_

11]1

,_

,_

f11. f(x) = (2x2 x + 3)(8x 3) + 4 f(x) 1 4x 894 3418 341412412+1]1

,_

11]1

+ ,_

,_

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f127 ()12. f(x) = 2x3 + kx2 + 5x + 4624 46 4 ) 2 ( 5 ) 2 ( ) 2 ( 26 ) 2 (2 3 + + +kkk f13. f(x) = 3x3 + 7x2 + kx 5927 34 5 ) 3 ( ) 3 ( 7 ) 3 ( 34 ) 3 (2 3 + + kkk f14. f(x) = kx4 + 30x3 + 18x + 209918113 20311831303113313 4 + ,_

+ ,_

+ ,_

,_

kkkf15. f(x) = 2x3 5x2 + 4kx 712 26 72142152126212 3 ,_

+ ,_

,_

,_

kkkf16. f(x) = x2 + ax (a + 2)12 2) 2 ( ) ( ) () (2 + + aa a a a a a a a f17. f(x) = px2 + 2x + 5345415449 45212215 ) 2 ( 2 ) 2 (21) 2 (22+ + + ,_

+ ,_

+ +

,_

ppppp pf f18. f(x) = x100 1 01 11 ) 1 () 1 (100 f19. f(x) = x1997 1 21 11 ) 1 () 1 (1997 f20. 23 2 13 ) 1 ( ) 1 ( 2 ) 1 (3 ) 1 (2 3 + + + + + k c c k c kf k = 1c = 3 k = 1c = 1 k = 3c = 1()21. f(x) = x2 4x 31 30 ) 1 )( 3 (0 3 46 3 ) ( 4 ) (6 ) (22 + + + + k kk k k kk kk f22. f(x) = x2 + 2x + 34 80 ) 4 )( 8 (0 32 411 322211222 + + + ,_

+ ,_

,_

k kk k k kk kkf23. f(x) = 4x2 2x + 110 ) 1 (0 1 23 1222432222 + + ,_

,_

,_

kk k kkk kkkf13 4B 24. f(x) = x3 + (a + 4)x2 2x 14310 ) 3 4 )( 1 (0 3 41 2 4 1 2 4 11 ) ( 2) )( 4 ( ) ( 1 ) 1 ( 2 ) 1 )( 4 ( ) 1 () ( ) 1 (22 3 32 3 2 3 + + + + + + + + + + + a aa a a aa a a a aa a a a aa f f25. f(x) = ax2 + bx 7 g(x) = 2x2 + ax + b f(x) x 3 ) 1 ( 7 321 3 914 7 ) 3 ( ) 3 (14 ) 3 (2 + + +b a b ab a f g(x) x + 3 ) 2 ( 17 31 ) 3 ( ) 3 ( 21 ) 3 (2 + + + b a b a g(1) (2) 424 6aa a = 4 (1)57 ) 4 ( 3 +bb26. (a) f(x) x 1 ) 1 ( 21 ) 1 ( ) 1 ( ) 1 ( 21 ) 1 (2 3 + + + q p q p f f(x) x + 2 ) 2 ( 11 231 ) 2 ( ) 2 ( ) 2 ( 231 ) 2 (2 3 + + + q p q p f(1) (2) 39 3pp p = 3 (1)52 3 +qq(b) (a) f(x) = 2x3 x2 + 3x 5 135 ) 2 ( 3 ) 2 ( ) 2 ( 2) 2 (2 3 + f27. (a) g(x) x 3 ) 1 ( 37 94 6 9 274 ) 3 ( 2 ) 3 ( ) 3 (4 ) 3 (2 + + + + + q p q p q pg g(x) 2x 1 ) 2 ( 248714 8187212212187212 + + + + ,_

+1]1

,_

,_

,_

qpqpq pg(1) (2) 435435 pp p = 4 (2)1244 + qq(b) (a) g(x) = x2 (x 4) + 2x 171 ) 4 ( 2 ) 4 4 ( ) 4 () 4 (2 + g28. f(x) = x2 mx + 3 g(x) = 2m x g(x) x n ) 1 ( 20 20 ) ( m nn m n g f(x) x n ) 2 ( 811 311 ) (22 + mn nmn n n f (1) (2)28 2 42 2t mm m m = 2 n = 4 m = 2 n = 4 m n '42nm ' 42nm29. (a) (x 1)(x + 2) 2 P(x) (x 1)(x + 2) 1147 ()(b) P(x) (x 1)(x + 2) Q(x) ax + b P(x) = (x 1)(x + 2) Q(x) + (ax + b) P(x) x 1 [ ]) 1 ( 44 ) 1 ( ) 1 ( ) 2 1 )( 1 1 (4 ) 1 ( + + + + b a b a Q P P(x) x + 2 [ ]) 2 ( 28 228 ) 2 ( ) 2 ( ) 2 2 )( 1 2 (28 ) 2 ( + + + + b a b a Q P(1) (2) 824 3aa a = 8 (1)124 8 +bb P(x) (x 1)(x + 2) 8x 1230. (a) f(x) = x99 + k21 11 ) 1 (1 ) 1 (99 + + kkkf(b) f(x) x + 1 Q(x)) 1 ( 1 ) ( ) 1 (1 ) ( ) 1 ( 21 ) ( ) 1 ( ) (9999 + + + ++ + x Q x x x Q x x x Q x x f x = 9 (1)9 ] 1 ) 9 ( [ 101 ) 9 ( 101 ) 9 ( ) 1 9 ( 999+ + QQ Q 999 10 9 7C (7.25 )1. (a)02 10 16 242 ) 2 ( 5 ) 2 ( 4 ) 2 ( 3 ) 2 (2 3+ + f(b) x 2 f(x) 2. (a)012727112172114218212 3 + ,_

+ ,_

,_

,_

g(b) g(x) 2x 1 3. (a)06 5 2 16 ) 1 ( 5 ) 1 ( 2 ) 1 ( ) 1 (2 3 + + + f x + 1 f(x) (b)0246 15 18 276 ) 3 ( 5 ) 3 ( 2 ) 3 ( ) 3 (2 3 + + f x 3 f(x) 4. (a)01915 12 80 6415 ) 4 ( 3 ) 4 ( 5 ) 4 ( ) 4 (2 3+ + + + + f x + 4 f(x) (b)015 15 125 12515 ) 5 ( 3 ) 5 ( 5 ) 5 ( ) 5 (2 3+ + + + + f x + 5 f(x) 5. (a)09 12 1 2921242142116212 3+ + + ,_

+ ,_

+ ,_

,_

f 2x + 1 f(x) (b)09 36 9 54923242342316232 3+ + + + ,_

+ ,_

+ ,_

,_

f 2x 3 f(x) 6. (a)0642516316164125413414412 3+ + + ,_

,_

+ ,_

,_

f 4x 1 f(x) (b)049964751627162764325433434432 3 + + + + ,_

,_

+ ,_

,_

f 4x + 3 f(x) 7. (a)02035291698203226324323322 3 + + ,_

+ ,_

+ ,_

,_

f 3x 2 f(x) (b)03164203104964964203426344343342 3 + ,_

+ ,_

+ ,_

,_

f 3x + 4 f(x) 15 4B 8. f(x) = x3 a3f(a) = a3 a3 = 0 x3 a3 a x 9. f(x) = 4x2 + 2(k 1)x k0) (2) 1 ( 22422 22 ,_

+ ,_

,_

k k k kkkkk kf 2x + k 4x2 + 2(k 1)x k 10. f(x) = 3x3 + kx2 8x 4 f(x) x 2 14 40 4 ) 2 ( 8 ) 2 ( ) 2 ( 30 ) 2 (2 3 +kkk f11. f(x) = 6x3 + 13x2 2bx + 3 x + 3 f(x) 742 60 3 ) 3 ( 2 ) 3 ( 13 ) 3 ( 60 ) 3 (2 3 + + bbb f12. f(x) = mx3 x2 7x + 6 2x 3 f(x) 24278270 623723230232 3 + ,_

,_

,_

,_

mmmf13. f(x) = 3x3 + 7x2 (5k + 2)x 2k f(x) 3x + 1 43430 2323597910 231) 2 5 (3173130312 3 + + + ,_

+ ,_

+ ,_

,_

kkkkk kf14. P(x) 2x + 3 ) 3 2 )( ( ) (2+ + + x c bx ax x P , ab c a 0 a = 1b = 0 c = 1 P(x)3 2 3 2) 3 2 )( 1 ( ) (2 32 + + x x x x x x P a = 1b = 1 c = 2 P(x)6 5 26 4 3 2 3 2) 3 2 )( 2 ( ) (2 32 2 32 + + + + + + x x x x x x x x x x x x P 3 2 3 22 3 + x x x 6 5 22 3 + x x x ()15. f(x) = x3 + ax2 x b x + 2 f(x) 6 40 ) 2 ( ) 2 ( ) 2 (0 ) 2 (2 3 + a bb af a = 3 b = 4(3) 6 = 18 a = 1 b = 4(1) 6 = 10 a = 1 b = 4(1) 6 = 2()16. f(x) = 2x2 + 5x 12 f(x) x + a 4230 ) 4 )( 3 2 (0 12 5 20 12 ) ( 5 ) ( 20 ) (22 + + a aa a a aa aa f17. f(x) = 15x2 (b + 3)x 1 f(x) 3x + b 2110 ) 1 2 )( 1 (0 1 20 13 9150 13) 3 (315032222 + + + + ,_

+ ,_

,_

b bb b b bbbbbbbbf18. (a) g(5) = (5)3 + 12(5)2 + 41(5) + 30 = 125 + 300 205 + 30 = 0 x + 5 g(x) 167 ()(b) 30 6 30 6 35 7 41 7 530 41 2 1 56 7 222 32 32++++++ + + ++ +xxx x x xx x x x x xx xg(x) = (x + 5)(x2 + 7x + 6)x2 + 7x + 6 = (x + 1)(x + 6)) 6 )( 1 )( 5 ( ) ( + + + x x x x g19. (a)0100 1009649641003475344343342 3 + + ,_

,_

+ ,_

,_

h 3x + 4 h(x) (b) 100 75 100 75 4 3100 75 4 3 4 325 2 32 32 + + +xxx x x x x xx) 5 )( 5 )( 4 3 () 25 )( 4 3 ( ) (2 + + + x x x x x x g20. f(x) = x3 4x2 + kx + 6(a) f(x) x 3 13 30 6 ) 3 ( ) 3 ( 4 ) 3 (0 ) 3 (2 3 + + kkk f(b) 6 2 6 2 3 36 4 32 222 32 32+ + + + + + xxx xx xx xx x x xx xf(x) = (x 3)(x2 x 2)x2 x 2 = (x 2)(x + 1)) 1 )( 2 )( 3 ( 6 42 3+ + + x x x x x x21. f(x) = 8x3 + mx2 25x + 6(a) f(x) 4x 1 281160 64125414180412 3 + ,_

,_

+ ,_

,_

mmmf(b) 6 24 6 24 4 25 4 2 86 25 2 8 1 46 2 222 32 32+ + + + +xxx x x xx x x x x xx xf(x) = (4x 1)(2x2 + x 6)2x2 + x 6 = (2x 3)(x + 2)) 2 )( 3 2 )( 1 4 ( 6 25 2 82 3+ + + x x x x x x22. f(x) = ax3 + 5x2 + bx 6 Q(x)) ( ) 1 )( 3 2 () ( ) 3 2 ( ) (2x Q x x x Q x x x f + + f(x) 2x + 3 x 1 023 ,_

f f(1) = 0) 1 ( 14 4 90421238270 623235230232 3 + + ,_

+ ,_

+ ,_

,_

b ab ab af) 2 ( 10 6 ) 1 ( ) 1 ( 5 ) 1 (0 ) 1 (2 3 + + +b ab a f(1) (2) 4 210 5aa a = 2 (2)11 2 +bb17 4B 23. f(x) = 2x3 + px2 25x + q Q(x)) ( ) 1 2 )( 4 () ( ) 4 7 2 ( ) (2x Q x x x Q x x x f + + f(x) x + 4 2x 1 f(4) = 0 021 ,_

f) 1 ( 28 160 ) 4 ( 25 ) 4 ( ) 4 ( 20 ) 4 (2 3 + + + q p q p f) 2 ( 49 40449402125212120212 3 + + + ,_

,_

+ ,_

,_

q pqpq pf(1) 4 (2) 163 63pp p = 1 (1)1228 ) 1 ( 16 +qq24. (a) f(x) x 4 ) 1 ( 148 160 ) 4 ( 5 ) 4 ( ) 4 ( 20 ) 4 (2 3 + + b a b a f 2x + 3 g(x) ) 2 ( 36 27 802982702315238230232 3 + + ,_

,_

,_

,_

b ab aa bg(1) (2) 2 4220 55bb b = 4 (1)9148 4 16 +aa(b) [ ]0116) 9 30 32 32 ( 24 10 36 169 ) 2 ( 15 ) 2 ( 8 ) 2 ( 4 2 4 ) 2 ( 5 ) 2 ( 9 ) 2 ( 2 ) 2 ( 2 ) 2 (2 32 3 + + + + + + + g f x + 2 ) ( 2 ) ( x g x f + 25. (a) x 2 f(x) h(x) f(2) = 0 h(2) = 0) 1 ( 5 20 10 2 40 2 ) 2 ( ) 2 ( ) 2 (0 ) 2 (2 3 + + + + + +q pq p q p f) 2 ( 1 20 2 2 40 6 ) 2 ( ) 2 ( ) 2 (0 ) 2 (2 3 + + + + +p qp q p q h(1) 2 (2) 39 3 pp p = 3 (1)15 ) 3 ( 2 + qq(b)2 10 ) 2 )( 1 (0 20 8 4 46 3 2 3) ( ) (222 3 2 3 + + + + x xx x x xx xx x x x x xx h x f26. (a) ) 1 ( 1310 ) 1 ( ) 1 ( ) 1 ( 210 ) 1 (2 3 + + + b a b a f x 1 f(x) ) 2 ( 10 ) 1 ( ) 1 ( ) 1 ( 20 ) 1 (2 3 + + + b a b a f(2) (1) 714 2 aa a = 7 (2)61 7 + bb(b) 6 6 6 6 7 2 26 7 2 16 2 222 32 32+ + + +xxx xx xx xx x x xx xf(x) = (x 1)(2x2 + x 6)2x2 + x 6 = (2x 3)(x + 2)) 2 )( 3 2 )( 1 ( ) ( + x x x x f187 ()27. (a) ) 1 ( 48 5 ) 1 ( ) 1 ( 9 ) 1 (8 ) 1 (2 3 + t st s g x + 5 g(x) ) 2 ( 44 250 5 ) 5 ( ) 5 ( 9 ) 5 (0 ) 5 (2 3 + t st s g(2) (1) 248 24ss s = 2 (1)64 2 tt(b) 5 5 5 6 10 25 6 9 2 51 2 222 32 32 + + + xxx xx xx xx x x xx xg(x) = (x + 5)(2x2 x 1)2x2 x 1 = (2x + 1)(x 1)) 1 2 )( 5 )( 1 ( ) ( + + x x x x g 7D (7.31 )1. f(x) = x3 4x2 + 5x 2 f(1) = 13 4(1)2 + 5(1) 2 = 0 x 1 f(x) x3 4x2 + 5x 2 = (x 1)(x2 3x + 2)2 2 3) 1 )( 2 ( 2 5 4 + x x x x x2. f(x) = x3 3x2 6x + 8 f(1) = 13 3(1)2 6(1) + 8 = 0 x 1 f(x) x3 3x2 6x + 8 = (x 1)(x2 2x 8)) 2 )( 1 )( 4 ( 8 6 32 3+ + x x x x x x3. f(x) = x3 + 3x2 4x 120 12 ) 2 ( 4 ) 2 ( 3 2 ) 2 (6 12 ) 1 ( 4 ) 1 ( 3 ) 1 ( ) 1 (12 12 ) 1 ( 4 ) 1 ( 3 1 ) 1 (2 32 32 3 + + + ff f x 2 f(x) x3 + 3x2 4x 12 = (x 2)(x2 + 5x + 6)) 3 )( 2 )( 2 ( 12 4 32 3+ + + x x x x x x4. f(x) = x3 + 8x2 + 21x + 180 18 ) 2 ( 21 ) 2 ( 8 ) 2 ( ) 2 (4 18 ) 1 ( 21 ) 1 ( 8 ) 1 ( ) 1 (2 32 3 + + + + + + ff x + 2 f(x) x3 + 8x2 + 21x + 18 = (x + 2)(x2 + 6x + 9)2 2 3) 3 )( 2 ( 18 21 8 + + + + + x x x x x5. f(x) = x3 9x2 + 15x + 250 25 ) 1 ( 15 ) 1 ( 9 ) 1 ( ) 1 (32 25 ) 1 ( 15 ) 1 ( 9 1 ) 1 (2 32 3 + + + + ff x + 1 f(x) x3 9x2 + 15x + 25 = (x + 1)(x2 10x + 25)2 2 3) 5 )( 1 ( 25 15 9 + + + x x x x x6. f(x) = x3 + 4x2 11x 300 30 ) 2 ( 11 ) 2 ( 4 ) 2 ( ) 2 (28 30 ) 2 ( 11 ) 2 ( 4 2 ) 2 (16 30 ) 1 ( 11 ) 1 ( 4 ) 1 ( ) 1 (36 30 ) 1 ( 11 ) 1 ( 4 1 ) 1 (2 32 32 32 3 + + + + f ff f x + 2 f(x) x3 + 4x2 11x 30 = (x + 2)(x2 + 2x 15)) 5 )( 2 )( 3 ( 30 11 42 3+ + + x x x x x x7. f(x) = x3 + 6x2 320 32 ) 2 ( 6 2 ) 2 (27 32 ) 1 ( 6 ) 1 ( ) 1 (25 32 ) 1 ( 6 1 ) 1 (2 32 32 3 + + + ff f x 2 f(x) x3 + 6x2 32 = (x 2)(x2 + 8x + 16)2 2 3) 4 )( 2 ( 32 6 + + x x x x8. f(x) = x3 + 10x2 + 33x + 360 36 ) 3 ( 33 ) 3 ( 10 ) 3 ( ) 3 (2 36 ) 2 ( 33 ) 2 ( 10 ) 2 ( ) 2 (12 36 ) 1 ( 33 ) 1 ( 10 ) 1 ( ) 1 (2 32 32 3 + + + + + + + + + fff x + 3 f(x) x3 + 10x2 + 33x + 36 = (x + 3)(x2 + 7x + 12)) 4 ( ) 3 ( 36 33 102 2 3+ + + + + x x x x x19 4B 9. 3x3 6x2 12x + 24 = 3(x3 2x2 4x + 8) f(x) = x3 2x2 4x + 80 8 ) 2 ( 4 ) 2 ( 2 2 ) 2 (9 8 ) 1 ( 4 ) 1 ( 2 ) 1 ( ) 1 (3 8 ) 1 ( 4 ) 1 ( 2 1 ) 1 (2 32 32 3 + + + ff f x 2 f(x) x3 2x2 4x + 8 = (x 2)(x2 4) x3 2x2 4x + 8 = (x 2)2(x + 2)) 2 ( ) 2 ( 3 24 12 6 32 2 3+ + x x x x x10. 5x3 + 20x2 + 5x 30 = 5(x3 + 4x2 + x 6) f(x) = x3 + 4x2 + x 6 f(1) = 13 + 4(1)2 + 1 6 = 0 x 1 f(x) x3 + 4x2 + x 6 = (x 1)(x2 + 5x + 6) x3 + 4x2 + x 6 = (x 1)(x + 2)(x + 3)) 3 )( 2 )( 1 ( 5 30 5 20 52 3+ + + + x x x x x x11. 5x3 65x 60 = 5(x3 13x 12) f(x) = x3 13x 120 12 ) 1 ( 13 ) 1 ( ) 1 (24 12 ) 1 ( 13 1 ) 1 (33 f f x + 1 f(x) x3 13x 12 = (x + 1)(x2 x 12) x3 13x 12 = (x + 1)(x 4)(x + 3)) 3 )( 1 )( 4 ( 5 60 65 53+ + x x x x x12. x3 10x2 31x 30 = (x3 + 10x2 + 31x + 30) f(x) = x3 + 10x2 + 31x + 300 30 ) 2 ( 31 ) 2 ( 10 ) 2 ( ) 2 (8 30 ) 1 ( 31 ) 1 ( 10 ) 1 ( ) 1 (2 32 3 + + + + + + ff x + 2 f(x) x3 + 10x2 + 31x + 30 = (x + 2)(x2 + 8x + 15) x3 + 10x2 + 31x + 30 = (x + 2)(x + 3)(x + 5)) 5 )( 3 )( 2 ( 30 31 102 3+ + + x x x x x x13. f(x) = x3 8x2 + 4x + 480 48 ) 2 ( 4 ) 2 ( 8 ) 2 ( ) 2 (32 48 ) 2 ( 4 ) 2 ( 8 2 ) 2 (35 48 ) 1 ( 4 ) 1 ( 8 ) 1 ( ) 1 (45 48 ) 1 ( 4 ) 1 ( 8 1 ) 1 (2 32 32 32 3 + + + + + + + + ffff x + 2 f(x) x3 8x2 + 4x + 48 = (x + 2)(x2 10x + 24)) 2 )( 4 )( 6 ( 48 4 82 3+ + + x x x x x x14. f(x) = x3 27x 540 54 ) 3 ( 27 ) 3 ( ) 3 (108 54 ) 3 ( 27 3 ) 3 (8 54 ) 2 ( 27 ) 2 ( ) 2 (100 54 ) 2 ( 27 2 ) 2 (28 54 ) 1 ( 27 ) 1 ( ) 1 (80 54 ) 1 ( 27 1 ) 1 (333333 f ff ff f x + 3 f(x) x3 27x 54 = (x + 3)(x2 3x 18)2 3) 3 )( 6 ( 54 27 + x x x x15. f(x) = x3 12x2 + 47x 600 60 ) 3 ( 47 ) 3 ( 12 3 ) 3 (210 60 ) 2 ( 47 ) 2 ( 12 ) 2 ( ) 2 (6 60 ) 2 ( 47 ) 2 ( 12 2 ) 2 (120 60 ) 1 ( 47 ) 1 ( 12 ) 1 ( ) 1 (24 60 ) 1 ( 47 ) 1 ( 12 1 ) 1 (2 32 32 32 32 3 + + + + + ff ff f x 3 f(x) x3 12x2 + 47x 60 = (x 3)(x2 9x + 20)) 3 )( 4 )( 5 ( 60 47 122 3 + x x x x x x16. f(x) = x3 x2 33x 630 63 ) 3 ( 33 ) 3 ( ) 3 ( ) 3 (144 63 ) 3 ( 33 3 3 ) 3 (32 63 ) 1 ( 33 ) 1 ( ) 1 ( ) 1 (96 63 ) 1 ( 33 1 1 ) 1 (2 32 32 32 3 f ff f x + 3 f(x) x3 x2 33x 63 = (x + 3)(x2 4x 21)) 7 ( ) 3 ( 63 332 2 3 + x x x x x17. f(x) = x3 + 3x2 25x 750 75 ) 3 ( 25 ) 3 ( 3 ) 3 ( ) 3 (96 75 ) 3 ( 25 ) 3 ( 3 3 ) 3 (48 75 ) 1 ( 25 ) 1 ( 3 ) 1 ( ) 1 (96 75 ) 1 ( 25 ) 1 ( 3 1 ) 1 (2 32 32 32 3 + + + + f ff f x + 3 f(x) x3 + 3x2 25x 75 = (x + 3)(x2 25)) 5 )( 3 )( 5 ( 75 25 32 3+ + + x x x x x x18. f(x) = x3 x2 44x + 840 84 ) 2 ( 44 2 2 ) 2 (126 84 ) 1 ( 44 ) 1 ( ) 1 ( ) 1 (40 84 ) 1 ( 44 1 1 ) 1 (2 32 32 3 + + + ff f x 2 f(x) x3 x2 44x + 84 = (x 2)(x2 + x 42)) 7 )( 6 )( 2 ( 84 442 3+ + x x x x x x207 ()19. f(x) = x3 5x2 57x 990 99 ) 3 ( 57 ) 3 ( 5 ) 3 ( ) 3 (288 99 ) 3 ( 57 ) 3 ( 5 3 ) 3 (48 99 ) 1 ( 57 ) 1 ( 5 ) 1 ( ) 1 (160 99 ) 1 ( 57 ) 1 ( 5 1 ) 1 (2 32 32 32 3 ffff x + 3 f(x) x3 5x2 57x 99 = (x + 3)(x2 8x 33)) 11 ( ) 3 ( 99 57 52 2 3 + x x x x x20. f(x) = x3 15x2 + 72x 1080 108 ) 3 ( 72 ) 3 ( 15 3 ) 3 (320 108 ) 2 ( 72 ) 2 ( 15 ) 2 ( ) 2 (16 108 ) 2 ( 72 ) 2 ( 15 2 ) 2 (196 108 ) 1 ( 72 ) 1 ( 15 ) 1 ( ) 1 (50 108 ) 1 ( 72 ) 1 ( 15 1 ) 1 (2 32 32 32 32 3 + + + + + ff ff f x 3 f(x) x3 15x2 + 72x 108 = (x 3)(x2 12x + 36)) 3 ( ) 6 ( 108 72 152 2 3 + x x x x x21. f(x) = 2x3 25x2 + 67x + 40040267425414021672125212212 3+ + ,_

+ ,_

,_

,_

f 2x + 1 f(x) 2x3 25x2 + 67x + 40 = (2x + 1)(x2 13x + 40)) 1 2 )( 5 )( 8 ( 40 67 25 22 3+ + + x x x x x x22. f(x) = 3x3 + 8x2 68x 48048313693298483268328323322 3 + + ,_

,_

+ ,_

,_

f 3x + 2 f(x) 3x3 + 8x2 68x 48 = (3x + 2)(x2 + 2x 24)) 2 3 )( 6 )( 4 ( 48 68 8 32 3+ + + x x x x x x23. f(x) = 5x3 6x2 29x + 60652925625165129516515512 3+ + ,_

,_

,_

,_

f 5x 1 f(x) 5x3 6x2 29x + 6 = (5x 1)(x2 x 6)) 2 )( 3 )( 1 5 ( 6 29 6 52 3+ + x x x x x x24. 30x3 + 64x2 + 6x 4 = 2(15x3 + 32x2 + 3x 2) f(x) = 15x3 + 32x2 + 3x 20 2 ) 2 ( 3 ) 2 ( 32 ) 2 ( 15 ) 2 (252 2 ) 2 ( 3 ) 2 ( 32 ) 2 ( 15 ) 2 (12 2 ) 1 ( 3 ) 1 ( 32 ) 1 ( 15 ) 1 (48 2 ) 1 ( 3 ) 1 ( 32 ) 1 ( 15 ) 1 (2 32 32 32 3 + + + + + + + + f ff f x + 2 f(x) 15x3 + 32x2 + 3x 2 = (x + 2)(15x2 + 2x 1) 15x3 + 32x2 + 3x 2 = (x + 2)(5x 1)(3x + 1)) 2 )( 1 3 )( 1 5 ( 2 4 6 64 302 3+ + + + x x x x x x 7 (7.35 )1.xx x x x xx8 128 0 12 43 32 3 2+ + = x 3 = x 82.15 12 6 3 6 4 2 2 2 23 2 0 26 2 222 32 32+ + + + + ++ xxx x x xx x x x x xx x = 6 22+ x x =15 3.3 2 2 23 0 2 3 2 2 32 33 42 3 42 3+ + + + x x x xx x x x x x xx x =2 3x x =3 4.3 2 4 1 4 2 4 2 4 2 41 2 6 4 1 22 2 2 222 32 32 + +++ + + +xxx x x xx x x x x xx x21 4B = 2 2 22 + x x =3 5. f(x)9 2 9 67 2 6 3 4 12 67 ) 2 6 3 )( 1 ( ) 2 6 3 )( 2 (7 ) 2 6 3 )( 1 2 ( ) (2 32 2 32 22+ + + + + + + + + + + + x x x x x x x x x x x x x x x x x f 6x3 9x2 2x + 96. p(x)5 270 3 5 25 2) 76 ( ) 6 3 5 2 () () 76 ( ) 5 2 ( ) ( 6 3 5 22 32 32 3++ + + + + + + x x x x x x x xx px x p x x x) 14 5 )( 5 2 ( 70 3 5 22 2 3+ + + + x x x x x x14 55 2) 14 5 )( 5 2 () (22+ ++ +x x x x x xx p x2 5x + 147. 18 ) 11 ( 3 23 15 5 5 6 2 2) 3 ( ) 5 )( 3 ( ) 2 )( 3 () 3 ( ) 5 2 )( 3 (7 3 22 32 2 32 222 3+ + + + + + + + + + + + + + x b x x bx x x x x x bx x x x x x bx x x x a x x xa = 18x 47 11 bb8. f(x) = 4x3 4x2 + x 1 01 ) 1 ( ) 1 ( 4 ) 1 ( 4) 1 (2 3 + f9. f(x) = x3 2x2 + 7x + 1 91 ) 1 ( 7 ) 1 ( 2 ) 1 () 1 (2 3 + + f10. f(x) = 16x3 2x + 1012122116213+ ,_

,_

,_

f11. f(x) = (x 699)699 (x 701)701 2) 1 ( 1) 1 ( 1) 701 700 ( ) 699 700 () 700 (21400701 699701 699

,_

ff 12. (a) f(1) = (1)4 + 29(1) + 6= 36 0 x 1 f(x) (b) f(3) = (3)4 + 29(3) + 6 = 0 x + 3 f(x) 13. (a) f(2)= (2)3 11(2)2 + 32(2) 28= 144 0 x + 2 f(x) (b) f(2) = (2)3 11(2)2 + 32(2) 28= 0 x 2 f(x) 14. (a)06321332133230322 3+ ,_

,_

,_

,_

f 3x + 2 f(x) (b)06211321132130212 3+ ,_

,_

,_

,_

f 1 2x f(x) 15. f(x) = x3 + kx2 5x + 6 f(x) x 2 14 40 6 ) 2 ( 5 ) 2 ( ) 2 (0 ) 2 (2 3 + +kkk f227 ()16. f(x) = kx3 3x2 8kx + 6 x + 3 f(x) 721 30 6 ) 3 ( 8 ) 3 ( 3 ) 3 (0 ) 3 (2 3 + kkk k f17. f(x) = 3x3 + 16x2 + x + c31953453131163135312 3 + + ,_

+ ,_

+ ,_

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cccf18. 91 1 1 3 4 2 11 1 ) 1 ( ) 1 ( 3 4 ) 1 ( ) 1 ( 2 1) 1 ( ) 1 (2 3 2 3+ + + + + + pppg f19. (a) f(x) = x3 x2 8x + 12012 24 9 2712 ) 3 ( 8 ) 3 ( ) 3 ( ) 3 (2 3+ + + f x + 3 x3 x2 8x + 12 (b) 22 2 3) 2 )( 3 () 4 4 )( 3 ( 12 8 + + + + x xx x x x x x20. f(x) = x3 2x2 7x 40 4 ) 1 ( 7 ) 1 ( 2 ) 1 ( ) 1 (12 4 ) 1 ( 7 ) 1 ( 2 1 ) 1 (2 32 3 f f x + 1 f(x) x3 2x2 7x 4 = (x + 1)(x2 3x 4)2 2 3) 1 )( 4 ( 4 7 2 + x x x x x21. f(x) = x3 + 5x2 2x 240 24 ) 2 ( 2 ) 2 ( 5 2 ) 2 (18 24 ) 1 ( 2 ) 1 ( 5 ) 1 ( ) 1 (20 24 ) 1 ( 2 ) 1 ( 5 1 ) 1 (2 32 32 3 + + + ff f x 2 f(x) x3 + 5x2 2x 24 = (x 2)(x2 + 7x + 12)) 4 )( 3 )( 2 ( 24 2 52 3+ + + x x x x x x22. 3x3 + 21x2 + 48x + 36 = 3(x3 + 7x2 + 16x + 12) f(x) = x3 + 7x2 + 16x + 120 12 ) 2 ( 16 ) 2 ( 7 ) 2 ( ) 2 (2 12 ) 1 ( 16 ) 1 ( 7 ) 1 ( ) 1 (2 32 3 + + + + + + ff x + 2 f(x) ) 3 ( ) 2 () 6 5 )( 2 ( 12 16 722 2 3+ + + + + + + +x xx x x x x x) 3 ( ) 2 ( 3 36 48 21 32 2 3+ + + + + x x x x x23. f(x) = x3 + 2ax b1 ) ( ) 3 ( ) ( + x Q x x f , Q(x) x b x Q x b x f + + ) ( ) 3 ( 1 ) ( f(x) 1 + b x 3 b 1 21 2 1 ) (33 + + + + ax x b b ax x b x f b x Q x x b x f+ + + + ] 1 ) ( )[ 3 () 3 ( 1 ) ( 4 ) 1 2 (3 1 2) 3 ( 1 ) (33 + + + + + + + x a x x b b ax x x b x f()24.8 4 4 8 4 4 4 4 24 3 6 1 24 222 32 3 2 + + + + xx x x x x x x x x x x xx =4 x =8 4 x25.9 8 4 4 8 5 4 8 2 2 45 2 6 4 1 24 2 222 32 3 2+ + + + +xx x x x x x x x x x x xx =4 2 x =9 8 x23 4B 26.52 48 60 48 12 8 0 12 15 12 38 15 0 3 5 412 3 222 32 3 2+ + + ++ + + + xx x x xx x x x x x x xx =12 3 + x =52 48 x27.4 8 0 4 4 4 2 0 4 4 0 24 0 2 2 04 2 222 32 32 3 42 3 4 22+ + + + + + + + + ++ xx x x x x x x x x x x x x x x x x x xx x = 4 22+ x x =4 + x28. (a) 1 34 111 3) 6 ( ) 2 11 () () 6 ( ) 1 3 ( ) ( 2 1122 322 32 2 3+ + ++ + + + +x x x x x x x x x xx px x x p x x x) 1 3 )( 4 ( 4 112 2 3+ + + + x x x x x x41 3) 1 3 )( 4 () (22+ + + +x x x x x xx p(b) f(x) = x3 +2x2 3x + 8 128 ) 4 ( 3 ) 4 ( 2 ) 4 () 4 (2 3 + + f29. ) 1 ( ) 5 ( ) 3 2 ( 63 1 3 2 2 6) 3 ( ) 1 )( 1 3 ( ) 2 )( 1 3 () 3 ( ) 1 2 )( 1 3 (5 13 62 32 2 32 222 3 + + + + + + + + + + + + + + + + + r x q x q x r x qx x x qx x r x qx x x qx x r x x qx x px x xx25) 3 2 ( 13+ q qx 10 5 + q p45 1 rr30. f(x) = 8x2 + 2x 1 2x k f(x) 2110 ) 1 2 )( 1 (0 1 20 122280222 + + ,_

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k kk k k kk kkf31. f(x) = x(x 1)(x 2) 2(x2 + 2x 12)(a) f(4) = (4)(4 1)(4 2) 2[(4)2 + 2(4) 12]= 0 x 4 f(x) (b)24 2 524 4 2 2 2) 12 2 ( 2 ) 2 )( () 12 2 ( 2 ) 2 )( 1 ( ) (2 32 2 2 32 22+ + + + + x x xx x x x x xx x x x xx x x x x x fx3 5x2 2x + 24 = (x 4)(x2 x 6)) 3 )( 2 )( 4 () 12 2 ( 2 ) 2 )( 1 (2 + + x x x x x x x x32. f(x) = x3 7x2 6x + 720 72 ) 3 ( 6 ) 3 ( 7 ) 3 ( ) 3 (18 72 ) 3 ( 6 ) 3 ( 7 3 ) 3 (48 72 ) 2 ( 6 ) 2 ( 7 ) 2 ( ) 2 (40 72 ) 2 ( 6 ) 2 ( 7 2 ) 2 (70 72 ) 1 ( 6 ) 1 ( 7 ) 1 ( ) 1 (60 72 ) 1 ( 6 ) 1 ( 7 1 ) 1 (2 32 32 32 32 32 3 + + + + + + f ff ff f x + 3 f(x) x3 7x2 6x + 72 = (x + 3)(x2 10x + 24)) 3 )( 4 )( 6 ( 72 6 72 3+ + x x x x x x33. 2x3 + 19x2 27x 90 = (2x3 19x2 + 27x + 90) f(x) = 2x3 19x2 + 27x + 900 90 ) 5 ( 27 ) 5 ( 19 ) 5 ( 2 ) 5 (216 90 ) 3 ( 27 ) 3 ( 19 ) 3 ( 2 ) 3 (54 90 ) 3 ( 27 ) 3 ( 19 ) 3 ( 2 ) 3 (56 90 ) 2 ( 27 ) 2 ( 19 ) 2 ( 2 ) 2 (84 90 ) 2 ( 27 ) 2 ( 19 ) 2 ( 2 ) 2 (42 90 ) 1 ( 27 ) 1 ( 19 ) 1 ( 2 ) 1 (100 90 ) 1 ( 27 ) 1 ( 19 ) 1 ( 2 ) 1 (2 32 32 32 32 32 32 3 + + + + + + + + + + + + + + fffffff x 5 f(x) 2x3 19x2 + 27x + 90 = (x 5)(2x2 9x 18) 2x3 19x2 + 27x + 90 = (x 5)(x 6)(2x + 3)) 3 2 )( 6 )( 5 ( 90 27 19 22 3+ + x x x x x x247 ()34. (a) f(2) = 2(2)4 + 11(2)3 5(2)2 98(2) 120 = 0 f(x) x + 2 (b) f(x) = (x + 2)(2x3 + 7x2 19x 60) g(x) = 2x3 + 7x2 19x 6010 60 ) 2 ( 19 ) 2 ( 7 ) 2 ( 2 ) 2 (54 60 ) 2 ( 19 ) 2 ( 7 ) 2 ( 2 ) 2 (36 60 ) 1 ( 19 ) 1 ( 7 ) 1 ( 2 ) 1 (70 60 ) 1 ( 19 ) 1 ( 7 ) 1 ( 2 ) 1 (2 32 32 32 3 + + + + ggggg(3) = 2(3)3 + 7(3)2 19(3) 60 = 0 g(x) x 3 ) 5 2 )( 4 )( 3 )( 2 () ( ) 2 ( ) () 5 2 )( 4 )( 3 () 20 13 2 )( 3 ( ) (2+ + + + + + + + x x x x x g x x f x x x x x x x g35. f(x) = 2x3 9x2 + px 6(a) 2x 3 f(x) 13239230 6232392320232 3 ,_

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pppf(b) ) 3 2 )( 1 )( 2 ( 6 13 9 2) 3 2 )( 2 3 ( 6 13 9 22 32 2 3 + + + x x x x x x x x x x x x36. f(x) = x3 2x2 + ax + b f(x) x + 1 ) 1 ( 58 ) 1 ( ) 1 ( 2 ) 1 (8 ) 1 (2 3 + + b a b a f f(x) x 2 ) 2 ( 4 24 ) 2 ( ) 2 ( 2 ) 2 (4 ) 2 (2 3 + + + b a b a f(1) + (2) 39 3aa a = 3 (1)25 3 bb37. f(x) = 2x3 x2 ax + b g(x) = bx3 2x2 x + a x 1 f(x) g(x) f(1) = 0 g(1) = 0) 1 ( 10 ) 1 ( ) 1 ( ) 1 ( 20 ) 1 (2 3 + b a b a f) 2 ( 30 1 ) 1 ( 2 ) 1 (0 ) 1 (2 3 + + b a a bg(1) + (2) 24 2aa a = 2 (1)11 2 bb38. f(x) = ax3 11x2 + bx + 40 Q(x)) ( ) 4 )( 2 () ( ) 8 2 ( ) (2x Q x x x Q x x x f + f(x) x + 2 x 4 f(2) = 0 f(4) = 0) 1 ( 2 40 40 2 44 80 40 ) 2 ( ) 2 ( 11 ) 2 (0 ) 2 (2 3 + + + + b ab a b a f) 2 ( 34 160 40 4 176 640 40 ) 4 ( ) 4 ( 11 ) 4 (0 ) 4 (2 3 + + + + + b ab a b a f(2) (1) 336 12aa a = 3 (1)142 ) 3 ( 4 +bb39. Q(x)) 2 3 ( ) ( ) 1 3 )( 4 () 2 3 ( ) ( ) 4 11 3 ( 19 62 2 3 + + + + + x x Q x x x x Q x x b ax x x x = 4[ ] [ ]) 1 ( 70 410 4 304 3842 ) 4 ( 3 ) 4 ( 1 ) 4 ( 3 ) 4 4 ( ) 4 ( ) 4 ( 19 ) 4 ( 62 3 + + + + + + + b a b a Q b a 31 x ) 2 ( 2 333 9192762313311313 4313131193162 3 + 1]1

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b abaQb a(1) 3 + (2) 16208 13 aa a = 16 (1)670 ) 16 ( 4 + bb25 4B 40. f(x) = 2x3 5x2 + kx + 6(a) f(x) x + 2 ) 1 ( 30 26 2 20 166 ) 2 ( ) 2 ( 5 ) 2 ( 2) 2 (1112 31 + + + + R k R k R k R f f(x) x 2 ) 2 ( 26 228 6 2 20 1628 6 ) 2 ( ) 2 ( 5 ) 2 ( 2) 2 (1112 32 + + + + + + R k R k R kR f(1) + (2) 14 4 kk(b) 0623235232232 3+ ,_

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f(c) (b) 2x 3 f(x) ) 2 )( 1 )( 3 2 ( 6 5 2) 2 )( 3 2 ( 6 5 22 32 2 3 + + + x x x x x x x x x x x x41. (a) f(3) = 4(3)3 + 3(3)2 + 5k(3) + 15 = 15k + 150g(3) = (3)3 + k(3)2 + 46(3) 21 = 9k + 901060 690 9 150 15) 3 ( ) 3 ( + +kk k k g f(b) f(3) = 15(10) + 150 = 0g(3) = 9(10) + 90 = 0 x 3 f(x) g(x) ) 6 )( 2 5 )( 3 () 12 28 5 )( 3 ()] 7 13 ( ) 5 15 4 )[( 3 () 7 13 )( 3 ( ) 5 15 4 )( 3 ( ) ( ) () 7 13 )( 3 ( ) () 5 15 4 )( 3 ( ) (2222222+ + + + + + + + x x x x x xx xx x xx x xx x x x g x fx x x x gx x x x f42. (a) = f(1 + 1) = f(0) = 03 + k = k1 kk x x k x x f + + + + + +2 3) 1 ( ) 1 ( ) 1 ( ) 1 ( f(x + 1) x + 1 k1 k(b) = f[2(2)] = f(4) = 43 + 1 65 43. (a) = f(2 1) = 4f(1) = 412 + a(1) + b = 4) 1 ( 3 a b (b) (i) f(x + 1) x + 2 ) 2 ( 10 ) 1 ( ) 1 (0 ) 1 (0 ) 1 2 (2 + + + a bb a ff(ii) (1) (2)21 3 a a a a = 2 (2)b = (2) 1 =144. f(x) = x99 + 1(a) 01 11 ) 1 () 1 (99+ + f(b) f(x) x + 1 Q(x)) 1 ( 1 ) ( ) 1 () ( ) 1 ( 1(a)) ( ) ( ) 1 ( ) (9999 + + ++ x Q x xx Q x xx Q x x f x = 6 (1)1 ) 6 ( 71 ) 6 ( ) 1 6 ( 699 + Q Q 699 45. (a) f(x) = x2016 1 01 11 ) 1 () 1 (2016 f(b) f(x) x 1 Q(x)) 1 ( 1 ) ( ) 1 () ( ) 1 ( 1(a)) ( ) ( ) 1 ( ) (20162016 + x Q x xx Q x xx Q x x f x = 25 (1)1 ) 25 ( 241 ) 25 ( ) 1 25 ( 252016+ + Q Q 18:00252016 19:00267 ()46. (a) 3 ) 1 ( 2 ) 1 () 1 (+ + nf03 2 1+ ( n )(b) f(x) x 1 63 ) 1 ( 2 1) 1 (+ + nf f(x) x2 1 Q(x) ax + b) 1 ( ) ( ) ( ) 1 )( 1 () ( ) ( ) 1 ( ) (2 b ax x Q x x b ax x Q x x f+ + + + + x = 1 (1)) 2 ( 6] ) 1 ( [ ) 1 ( ) 1 1 )( 1 1 ( ) 1 ( b a b a Q f+ + + + x = 1 (1)) 3 ( 0] ) 1 ( [ ) 1 ( ) 1 1 )( 1 1 ( ) 1 ( b a b a Q f+ + + + (2) + (3) 36 2bb b = 3 (3)33 0+ a a f(x) x2 1 3x + 347. (a) f(x) = 2x2 + px + q g(x) = 2x2 + qx + p x r f(x) g(x) q p r q p p qr r q pr r r g r f + + + +) (2 2) ( ) (2 21 r( p q )(b) (a) 2x2 5x + 3 (x 1)(x + 5) x2 + 4x 5 () (7.38 )1. A f(x) = x3 x2 + 2x + 191 4 4 81 ) 2 ( 2 2 2) 2 (2 3+ + + + f2. C f(x) x + 1 00 1 10 ) 1 ( ) 1 (0 ) 1 (2009 + + + kkkf3. D f(1) = 040 1 30 1 ) 1 ( ) 1 ( 32 + + + ppp1 4 3 ) (2+ + x x x f81 4 31 ) 1 ( 4 ) 1 ( 3 ) 1 (2+ + + + f4. A f(1) = 0 034 ,_

f x 1 3x + 4 f(x) A5. C) ( 2 242 62 ) 3 )( 2 () (22 + + + + kkk k kk k k k k f6. A P(x) 4x 1 Q(x)R x Q x R x Q x x P+ + )] ( )[ 4 1 () ( ) 1 4 ( ) ( P(x) 1 4x R7. B Q(x) x + 1 Q(x 1) (x 1) + 1 = x 8. A f(x) = 2x3 ax2 + bx + 3 x + 3 3 22 3+ + bx ax x 17 30 3 3 9 540 3 ) 3 ( ) 3 ( ) 3 ( 20 ) 3 (2 3 + + + b ab ab a f9. D f(x) x + 2 2x 1 0 ) 2 ( f 021 ,_

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b ab af (2) (1)840 56 ) 17 2 ( 2 bbb b b = 8 (1)16 ) 8 ( 2 aa10. B p(x) x2 1 Q(x) ax + b......(1) ) ( ) ( ) 1 )( 1 () ( ) ( ) 1 ( ) (2b ax x Q x x b ax x Q x x p+ + + + + p(x) x 1 1......(2) 11 ] ) 1 ( [ ) 1 ( ) 1 1 )( 1 1 ( ) 1 ( + + + + b a b a Q p p(x) x + 1 3......(3) 33 ] ) 1 ( [ ) 1 ( ) 1 1 )( 1 1 ( ) 1 ( + + + + b a b a Q p(2) + (3) 2b = 4b = 2 b = 2 (2)a + 2 = 1a = 1 p(x) x2 1 x + 2 (7.39 ) f(x) = x3 + kx2 + 3324 824 92 3 ) 1 ( ) 1 ( 3 ) 3 ( ) 3 (2 ) 1 ( ) 3 (2 3 2 3 + + + + kk k k k k f f (7.40 )1. (a) (i)) 1 )( 1 ( 1) 1 )( 1 ( 1) 1 )( 1 )( 1 () 1 )( 1 ( 122 322 2 4 + + + + + + + x x xx x x xx x x x x x(ii) (a)(i) x + 1 x4 1x3 + 1 x2 1 (b) (i)) 1 ( ) 1 () 1 ( ) 1 () () (23 42 3 4+ + + + + + + + + + + x D x C x B x A E D C B A E Dx Cx Bx Ax F x f (a)(i) x + 1 x4 1x3 + 1 x2 1 x + 1 f(x) F (ii) (b)(i) ) ( ) 1 ( ) ( x Q x F x f + Q(x) x F Q f Q F f+ + ) 10 ( 11 ) 10 () 10 ( ) 1 10 ( ) 10 ( F 11 F = 11M M ] ) 10 ( [ 1111 ) 10 ( 11 ) 10 (M Q M Q f+ + f(10) 11 2. ABCDE f(10) = A 104 + B 103 + C 102 + D 10 + E= ABCDE 1(b)(ii) A B + C D + E 11 11 3. 31 658A = 3B = 1C = 6D = 5 E = 8F = A B + C D + E= 3 1 + 6 5 + 8= 11 F 11 31 658 11 35 618A = 3B = 5C = 6D = 1 E = 8F = A B + C D + E= 3 5 + 6 1 + 8= 11 F 11 35 618 11 61 358A = 6B = 1C = 3D = 5 E = 8F = A B + C D + E= 6 1 + 3 5 + 8= 11 F 11 61 358 11 31 658 35 618 61 358 11 (ABCDE (B = 1 D = 5) (B = 5 D = 1))28