III 2017
0 7 1 2 11 2 1 2.1............................... 1 2.2.................................. 16 n 15.1 n................................ 15.2 de Moivre............................. 15. 1 n................................. 16.4 n.............................. 17.5 n................................ 17 4 4 19 4.1............................... 19 4.2.................................. 22 5 4 (2) 2 5.1............................... 2 5.2.................................. 25 6 2, 27 6.1 2 x 1 x 2......................... 27 6.2 x 1 + ωx 2 + ω 2 x.................... 27 7 1 7.1 n................................. 1 7.2................................. 2 7.................................... 4
4 8 9 8.1..................................... 9 8.2.................................. 9 8.................................. 41 9 45 9.1...................................... 45 9.2.................................. 46 9.................................... 46 9.4............................. 47 9.5.................................. 48 9.6................................ 49 10 51 10.1 2.............................. 51 10.2.............................. 52 10. 4.............................. 5 11 55
5 0 1 x a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 a n 0 ( ) a 0,..., a n n n ( ) ( ) x 1 1 1 1 2,, 4, 5,... *1 1 1 *1
6 0 1 2 1 2 *2 (aha) 1 7 19 * x + 1 7 x = 19. 1 2 4000 *4 1 870 1 1 1 2 1 2 1 2 = 1 4 870 870 + 1 4 = 481 4 1 2 2 59 2 59 2 + 1 = 0 1 2 2 x 2 x = 870 ( ) 2 ( ) 2 ( ) 2 1 1 59 + = 870 + =, 2 2 2 ( ) 2 ( ) 2 ( 1 1 + = x 2 x + = x 1 2, 2 2 2) *2 B. C. 1650 * *4 60
7 x 1 2 = 59 2, x = 59 2 + 1 2 = 0 2 0 2 9 al-khwārizmī 780 850 *5 hisb al-jabr wa l muqbala *6 1 2 jabr 6 bx = c. bx = ax 2. ax 2 = c. ax 2 + bx = c. bx + c = ax 2. ax 2 + c = bx. 2 5, 4 4 16 1545 Hieronimo Cardano, 1501-1576 Ars Magna *7 4 * 8 Ludovico Ferrari, 1522-1565 *5 *6 al-jabr algebra *7 *8
8 0, 4 5 5 4 5 *9 19 Niels Henrik Abel, 1802-1829 5 Evariste Galois, 1811-182 Julius Wilbelm Richard Dedkind, 181-1916 *9
9 1 2 x 2 ax 2 + bx + c = 0 a b c a 0 (1.1) (1.1) a x 2 + b a x + c a = 0. b2 4a 2 c a = x 2 + b a x + c a + b2 4a 2 c a = x 2 + b a x + + b2 4a ( 2 = x + b ) 2. 2a = b2 4a 2 c a = b2 4ac 4a 2. x + b 2a = ± b2 4ac b2 4ac 4a 2 = ± 2a x = b 2a ± b2 4ac. 2a 1.1. i 1 (1) x 2 2x 1 = 0. (2) x 2 + x + 1 = 0. () x 2 (1 + i)x + i = 0. (4) x 2 2x (2 + 4i) = 0.
10 1 2 2 (1.1) y y 2 = b2 4ac 4a 2 (1.2) 2 = b2 4ac 4a 2 (1.1) x = b 2a ±. (1.1) 2 x 1, x 2 2 = (x 1 x 2 ) 2 1.2.
11 2 a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 a n 0 ( ) a 0, a 1,..., a n i 1 ( ) a n x n + a n 1 x n 1 + + a 1 x + a 0 2.1 a 0 ax + bx 2 + cx + d = 0 ( ) 2.1. ( ) α, β a(x + α) = β. ( ) 1 x + ax 2 + bx + c = 0. ( ) y = x + a/ y + py + q = 0 ( ) ( ) 2 ( T 2 + qt + p = 0 ( ) )
12 2 2 U, V u, v u = U, v = V, uv = p u + v, ωu + ω 2 v, ω 2 u + ωv ( ) ω 1 ω ( ) ( ) ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) f(x) q = f( a/), p = f ( a/). 2.1. x + x 2 + 9x + 5 = 0. x = 1 + 4 + 2, 1 1 2 ( 4 + 2) ± 2 ( 4 ω 2)i. 2.2. x + 6x 20 = 0. 10 + 108 + 10 108, ω 10 + 108 + ω 10 2 108, ω 2 10 + 108 + ω 10 108 2 = (1 + ) + (1 ), 1 ± i
2.1 1 2.2. ω(1 + ) + ω 2 (1 ) = 1 + i 2.. x + 2x 4i = 0. T 2 4iT 8 27 = 0 ( ±10 + 6 ) T = i. u = (1 + )i = ( + )i ( 10 + 6 ) i v = 2 1 u = 2 ( ) ( + )i uv = 2, ( v 10 + 6 ) = i, u + v = 4i = ( )i u + v = ( + )i ( )i = 2i 2.. 2 2.4. x 4 x 1 8 = 0. T 2 1 8 + 1 64 = 0. T = 1 16 (1 ± i).
14 2 1/4 u, v u + v, ωu + ω 2 u, ω 2 + ωv. cos 40, cos 100, cos 140 2.2 (1) x + 6x 2 + x + 2 = 0. (2) x 9x 2 + 6x 48 = 0. () x + 2x 2 = 4 2 + 9. (4) x x 6 = 0. (5) x + 9x = 4 2. (6) x + 10x = 6x 2 + 4. (7) x + 21x = 9x 2 + 5. (8) x 9x + 2 2 = 0.. (1) (2) 1 2