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Transcription:

II

16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369

Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 + 4 = 2 P ( 1) = ( 1) 3 7 ( 1) + 4 = 1 + 7 + 4 = 10 P (a) = a 3 7a + 4 P (b + 1) = (b + 1) 3 7(b + 1) + 4 = b 3 + 3b 2 4b 2 3 ( ) a b + 1 a x a b + 1 171 P (x) = x 3 + 3x 2 x + 4 (1) P (1) (2) P ( 2) (3) P ( b) (4) P (a + 1) 1 P f polynomial function P, Q f, g, h 370

16.1.2 2 x α 1 2 3 ( ) P (x) x α P (α) P (x) x α Q(x) R 4 P (x) = (x α)q(x) + R P (α) = (α α)q(α) + R = 0 Q(α) + R = R P (x) x α P (α) α x P α x + α x + α = x ( α) α P (x) = x 3 4x + 1 x 2 P (2) = 2 3 4 2 + 1 = 1 1 P (x) x + 1 P ( 1) = ( 1) 3 4 ( 1) + 1 = 4 4 ( ) 2 α a a 3 4 1 x α 0 0 R 371

96 P (x) x 2 x + 1 1 4 172 P (x) = x 3 3x 2 + 4x 2 x + 1, x 2, x + 3 55 Q(x) = x 3 + 2x 2 + ax + 4 x + 1 12 a Q(x) x + 1 Q( 1) a 12 a Q(x) x + 1 Q( 1) = 1 + 2 a + 4 = a + 5 12 a + 5 = 12 a = 7 ( ) ( ) 173 P (x) = x 3 + x 2 + ax 2 x 2 8 a 56 P (x) x 2 4 x 3 6 P (x) (x 2)(x 3) 372

1 2 P (x) x 2 4 x 3 6 P (2) = 4, P (3) = 6 P (x) (x 2)(x 3) (x 2)(x 3) ax + b ( a, b ) a b 5 P (x) (x 2)(x 3) ( ) Q(x) P (x) = (x 2)(x 3)Q(x) + ax + b x 2 ( 2 x 2 2 ) P (2) = (2 2)(2 3)Q(2) + 2a + b (2 2)(2 3)Q(2) 2 2 = 0 0 ( 1) Q(2) = 0 P (x) x 2 4 P (2) = 4 4 = 2a + b (1) 6 x = 3 6 = 3a + b (2) a, b (1) (2) a, b 5 ax + b 1 a = 0 a = 0, b = 0 6 Q(x) 373

P (x) (x 2)(x 3) Q(x) ax + b P (x) = (x 2)(x 3)Q(x) + ax + b P (x) x 2 4 x 3 6 P (2) = 4, P (3) = 6 { 2a + b = 4 3a + b = 6 a = 2, b = 0 2x ( ) ( ) 174 x P (x) x 2 1 x + 3 9 P (x) (x 2)(x + 3) x α 1 7 ( ) P (x) 1 ax + b ( P b ) a 97 175 S(x) = x 3 + 2x 2 x 3 2x 1 7 a, b 1, α 374

16.1.3 P (x) x α 0 8 ( ) P (x) x α P (α) = 0 P (α) = 0 P (x) x α P (x) x α P (α) P (α) = 0 0 P (x) x α 98 P (x) x α P (α) = 0 x α x α P (x) = x 3 3x 2 P (2) = 2 3 3 2 2 = 0 P (x) x 2 P (1) = 1 3 3 1 2 = 4 P (x) x 1 ( ) 176 x 1, x + 1, x + 2, x 3 P (x) = x 4 + 2x 3 x 2 57 P (x) = x 3 5x 2 + 2x + 8 8 375

1 0 P (x) x α P (x) = (x α)(a 0 x n + + a n ) P (x) = a 0 x n+1 + αa n αa n 8 αa n = 8 α 8 9 α ±1, ±2, ±4, ±8 0 0 1 P ( 1) = ( 1) 3 5 ( 1) 2 + 2 ( 1) + 8 = 1 5 2 + 8 = 0 P (x) x + 1 9 a 0 x n + + a n α 0 376

P (x) x + 1 10 P (x) = (x + 1)(x 2 6x + 8) x 2 6x + 8 (x 2)(x 4) x 3 5x 2 + 2x + 8 = (x + 1)(x 2)(x 4) x 3 5x 2 + 2x + 8 = (x + 1)(x 2)(x 4) ( ) 177 (1) x 3 7x + 6 (2) x 3 2x 2 5x + 6 (3) x 3 6x 2 + 12x 8 (4) x 4 + 3x 3 5x 2 3x + 4 (5) x 4 x 3 7x 2 + x + 6 1 a 3 b 3 = (a b)(a 2 + ab + b 2 ) 10 377

a 3 b 3 a P (a) = a 3 b 3 11 0 a b P (b) = b 3 b 3 = 0 P (a) a b a 3 b 3 a b a 3 b 3 16.2 ( ) A B A = BQ B A A B x 3 5x 2 + 2x + 8 = (x + 1)(x 2 6x + 8) x + 1 x 3 5x 2 + 2x + 8 x 3 5x 2 + 2x + 8 x + 1 ( ) 2 11 2 P (x) x P (a) x a, b 378

16.3 16.3.1 ( ) 2 ( ) 2 a 2 bx 2 y 3 a 2 b 3 xy a 2 bxy a 2 b 3 x 2 y 3 ( ) 58 x 3 3x 2 + 2x x 4 10x 2 + 9 x 3 3x 2 + 2x = x(x 1)(x 2) x 4 10x 2 + 9 = (x + 1)(x 1)(x + 3)(x 3) 379

x 1 x(x + 1)(x 1)(x 2)(x + 3)(x 3) ( ) ( ) 178 (1) b 2, b 3 c 2 (2) (x + 1)(x 2), (x + 2)(x 2) (3) (x + 1)(x 2) 2, (x 2) 3 (x 1) 2 (4) x 2 + 6x + 8, x 2 + 5x + 6 (5) x 2 + x 6, x 2 + 4x + 3, x 2 + 3x (6) x 3 + 4x 2 3x 18, x 3 + 4x 2 9x 36 16.3.2 ( ) (0 ) 1 0 ( ) A, B A B Q R A B B R 99 1 59 x 3 3x 2 + 2x x 4 10x 2 + 9 380

x 4 10x 2 + 9 x 3 3x 2 + 2x 3x 2 6x + 9 x 3 3x 2 + 2x 3x 2 6x + 9 3x 2 x 3 1 3 x 1 3 x + 5 3 15x 15 3x 2 6x + 9 15x 15 ) 15x 15 15x 15 = 15(x 1) 15 x 1 ( ) 16.3.3 ( ) ( ) ( ) A, B D L AB = DL 381

( ) A, B D A = A D, B = B D A B A B D 100 60 2 x + 2 2x 3 + 3x 2 2x A, B A = A (x + 2), B = B (x + 2) A B A B (x + 2) x + 2 A, B A, B 2 A B 1 A, B A = A (x + 2), B = B (x + 2) A B 2x 3 + 3x 2 2x = x(x + 2)(2x 1) A, B 2 A, B 1 A = x, B = 2x 1 x(x + 2) (2x 1)(x + 2) ( ) ( ) A B A B A B A, B 179 P, Q 2 3 1 P, Q x+1 x 4 +3x 3 +4x 2 +3x+1 P, Q 382

16.4 (1976) 1 (2000) 1 2 383

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