Промежуточный контроль.
Многочлены от одной переменной.Операции над многочленами
Разделите с остатком многочлен f(x) на двучлен g(x) = (х - с), если:
1) Пример решения
а) f(x) = 3х6 + 2x5 - x4 + 3x3 + x - 3, g(x) = x - 1;
б) f(x) = iх5 +(i - 1)x4 + ix3 + (1 + 2i)x + 2, g(x) = x - 1 - i;
2)
а) f(x) = х8 - 2x7 + x6 + 2x5 - 4x4 + 3x3 - x2 + 2x - 1, g(x) = x - 2;
б) f(x) = х6 - 3x5 +(3 + i)x4 + 3ix3 - 4ix2 +(4 + 3i)x + i, g(x) = x - 1 - i;
3)
а) f(x) = 2х6 - 3x4 + 4x2 - 2x + 1, g(x) = x + 2;
б) f(x) = х6 -2ix5 + (1 -i)x4 + 2x3 + ix2 + 1, g(x) = x - i;
4)
а) f(x) = 2х8 - x6 - 3x4 - 2x + 3, g(x) = x + 3;
б) f(x) = х6 + ix5 - (1 + 2i)x4 + x2 - 1 - 3i, g(x) = x - 2 + i;
5)
а) f(x) = 3х7 - x6 + 3x3 - 2x + 5, g(x) = x + 1;
б) f(x)=х6+(1-i)x5+2ix4+(3+3i)x3+x2+(2i-1)x-2+3i, g(x) = i -1;
6)
а) f(x) = 2х5 - 5x3 - 3x2 + 4, g(x) = x - 2;
б) f(x) = (1 - i)х5- ix4+ix3 +(3 - 5i)x2 +(1 -2i)x - 4+3i, g(x) = x - 1 + i;
7)
а) f(x) = 4х7 - 3x4 + 2x2 - 3x - 3, g(x) = x + 2;
б) f(x) = iх5+ (1 - i)x4 + (-2 + i)x3 + ix + 3 - i, g(x) = x - 1 + i;
8)
а) f(x) = 2х5 + 4x4 - 5x3 + 3x2 - 2, g(x) = x + 2;
б) f(x) = (1 + i)х5+ ix4 + (1 - 2i)x3 + (i -1)x2 - 9 + 8i, g(x) = x - 2 + i;
9)
а) f(x) = 3х5 - 2x4 + x2 - 5, g(x) = x - 4;
б) f(x) = iх5+ (i - 2)x4 + (1 + i)x3 + ix + 4 + i, g(x) = x - i;
10)
а) f(x) = 2х7 + x5 - 3x3 + x - 4x, g(x) = x + 3;
б) f(x) = (3 + i)x6 - (2 - i)x4 + 3x3 - ix2 + 3, g(x) = x + 1 - 2i;
11)
а) f(x) = 2х5 + 5x4 - 2x3 + 2x2 + 5x + 20, g(x) = x + 3;
б) f(x) = (2 + 3i)x5 - 2(4 + 5i)x4 - 15ix3 + (6 + 13i)x2 + 8(3 + 2i)x + 55 + 18i, g(x) = x - 3 + 2i;
12)
а) f(x) = 4х5 - 5x3 + 2x2 - 8x, g(x) = x + 3;
б) f(x) = 3x5 - ix4 + (2 - 3i)x3 - 7x + 1 - 2i, g(x) = x - 1 + 2i;
13)
а) f(x) = 2х5 + 2x3 + x2 - 4x + 1, g(x) = x + 2;
б) f(x) = ix5 + (2 - i)x4 + (2 - 3i)x3 + ix2 - 7 + 2i, g(x) = x - 3 - 2i;
14)
а) f(x) = 3х7 + x6 + 2x5 - 5x4 + x3 + 3x2 - 7, g(x) = x - 1;
б) f(x) = ix5 + (1 + 2i)x4 + (1 - i)x3 + ix2 + (1 + i)x + 3, g(x) = x - 1 + i;
15)
а) f(x) = 4х6 - 3x5 + 2x3 - 6x2 + x - 3, g(x) = x + 3;
б) f(x) = ix6 - (2 + i)x4 + 3x2 - 2x - 7 + i, g(x) = x - 3i;
16)
а) f(x) = х5 - 3x4 + 5x3 + 4x2 + 2x - 10, g(x) = x - 1;
б) f(x) = (-1 + 2)x5 + (3 + i)x4 + (1 + 2i)x3 - (1 + i)x2 + (4 + i)x + 4 + i, g(x) = x - 1 - i;
17)
а) f(x) = 2х5 + 3x4 + 4x3 + x2 - 1, g(x) = x - 1;
б) f(x) = x5 + (1 + 2i)x4 - (1 + 3i)x2 + 7, g(x) = x - i;
18)
а) f(x) = х5 + 2x4 - 3x3 + 2x2 - x - 1, g(x) = x - 2;
б) f(x) = ix5 + ix4 - 2x3 + x2 + (1 - i)x + 4, g(x) = x - 1 + i;
19)
а) f(x) = 2х6 - 3x4 + 2x3 - x2 - 4x - 3, g(x) = x - 3;
б) f(x) = ix5 +(1 + i)x4 + (5 + i)x3 - (1 - i)x2 + (3 + i)x + i + 5, g(x) = x + 1 - i;
20)
а) f(x) = 2х6 + 4x3 - 4x2 + 2x - 4, g(x) = x + 1;
б) f(x) = x5 + ix4 - (-1 + i)x2 + 2 + i, g(x) = x - 1 + 2i;
Найдите значение многочлена f(x) при х = х0, если:
1) пример решения
f(x) = 2х6 - x5 - 3x4 - 2x2 + 4x - 3, x0 = -2;
2)
f(x) = 4х7 - 3x6 + 2x4 - 3x3 - 4x + 2, x0 = 2;
3)
f(x) = х8 - 3x7 + 4x4 + 2x3 - 3x2 + х - 1, x0 = 3;
4)
f(x) = iх6 + 2x5 - (2 + 3i)x3 + x2 - ix + 5, x0 = -i;
5)
f(x) = 2х7 - 3x6 + ix5 + (2 - i)x3 - 3x2 + ix - i, x0 = 1 - i;
6)
f(x) = х6 + ix5 + (2 - i)x4 - ix3 + (2 + 3i)x2 + 3ix + 1, x0 = 1 + i;
7)
f(x) = 3х7 - 2x5 + 6x3 - 4x2 - 3x + 1, x0 = -3;
8)
f(x) = iх6 + (3 - i)x5 + (2 - 3i)x4 + ix3 + (1 + i)x2 - 3ix + 6, x0 = i - 1;
9)
f(x) = 2х5 - x4 + 2x3 + x + 2 + 3i, x0 = 1 + 4i;
10)
f(x) = х5 + (2 - 2i)x4 - ix3 + 4x2 - ix + 2 - i, x0 = 2 + i;
11)
f(x) = 4х5 -(2 + 5i)x4 +(4 - 3i)x3 +(3 - i)x2 +(6 + 7i)x + 7 - 2i, x0 = 1 + 2i;
12)
f(x) = х5 + (1 + 2i)x4 - (1 + 3i)x3 + 7x - 6 + i, x0 = -2 - i;
13)
f(x) = 2х5 + (1 + i)x4 - ix3 + 2x2 + 2, x0 = 4 + i;
14)
f(x) = 7х8 - 2x5 + 4x4 - 3x3 + 4x - 10 + 4i, x0 = -5;
15)
f(x) = 2х9 - x7 - x3 + 2x2 - 1, x0 = 2i;
16)
f(x) = 2х8 - ix6 - ix5 + 2x3 - x - i, x0 = -2i;
17)
f(x) = х9 - x7 - 3x6 - 4x5 + 2x2 - 4, x0 = -2;
18)
f(x) = 3х6 - 4x5 - ix3 + ix2 + 7x + i, x0 = 1 + i;
19)
f(x) = 4х6 - ix4 + 2x2 - (3 + i)x - 4, x0 = 2 - i;
20)
f(x) = iх5 + 2x4 - (3 + i)x2 - 4x - 2 + i, x0 = 2 + i;
Проверьте, является ли число х0 = с корнем многочлена f(x).
1) пример решения
f(x) = 2х6 - 4x5 - 3x3 - 4x2 + 2x + 2, x0 = 4;
2)
f(x) = 3х6 - 8x5 - 7x3 + 4x2 + 3x + 6, x0 = 3;
3)
f(x) = 3х5 - 2x4 + x3 - x2 + x - 585, x0 = 3;
4)
f(x) = 2х6 - x5 + 3x4 - 2x3 - x2 + 3х + 42, x0 = -4;
5)
f(x) = 3х6 - 4x5 + 5x3 + x - 4, x0 = 2;
6)
f(x) = 2х5 - x4 + ix3 + (3 - 2i)x2 + 6 + 4i, x0 = 2i;
7)
f(x) = 3х5 + x4 - 2x2 + x - 106, x0 = 2;
8)
f(x) = 2х6 - 22x5 + 87x4 - 133x3 + 9x2 + 135x - 54, x0 = 3;
9)
f(x) = 2х6 - 3x5 + x4 - 2x3 + 4x2 - 8x + 6, x0 = 1;
10)
f(x) = х6 - 4x5 - 2x4 + 2x3 + 8x - 5, x0 = 1;
11)
f(x) = 5х6 - 7x5 - 6x4 + x3 - 3x2 + 3x - 2, x0 = 2;
12)
f(x) = 3х6 - 2x5 + 8x4 - 2x3 + 2x2 - 7x + 6, x0 = 1;
13)
f(x) = х6 - 2x5 - 15x4 + 18x3 + 46x2 - 27x + 12, x0 = 4;
14)
f(x) = х6 - 2x5 - 6x4 + 16x3 - 3x2 - 4x - 2, x0 = 2;
15)
f(x) = х6 - 5x5 + 10x4 - 4x3 + 14x2 + 28x - 20, x0 = 2;
16)
f(x) = х6 - 5x5 + 3x4 - 2x3 + 4x2 + x + 46, x0 = 2;
17)
f(x) = х6 - 5x5 + 6x4 + 2x3 - 7x2 - 6x - 9, x0 = 3;
18)
f(x) = 2х6 + 3x5 - 11x4 - 11x3 + 19x2 - 12x - 12, x0 = 2;
19)
f(x) = х6 - 2x5 + 3x3 - 12x2 + 2x - 1, x0 = 2;
20)
f(x) = -х6 + 2x5 - 2x4 + 3x3 + 8x2 - 11x + 2, x0 = 2;
С помощью схемы Горнера установите кратность корня х0 = с многочлена f(x).
1) пример решения
f(x) = x5 - 2x4 + 4x3 - 28x2 + 48x - 16, x0 = 2;
2)
f(x) = x4 - 2x3 + 5x2 - 28x + 36, x0 = 2;
3)
f(x) = x5 - 3x4 - x3 + 11x2 - 12x + 4, x0 = 1;
4)
f(x) = x5 - 6x4 + 13x3 - 14x2 + 12x - 8, x0 = 2;
5)
f(x) = x4 + 8x3 - x2 - 68x - 84, x0 = -2;
6)
f(x) = x5 - 6x4 + 13x3 - 14x2 + 12x - 8, x0 = 2;
7)
f(x) = x4 - 2x3 + 2x2 - 2x + 1, x0 = 1;
8)
f(x) = x5 - 6x4 + 11x3 - 2x2 - 12x + 8, x0 = 2;
9)
f(x) = x5 - 8x4 + 21x3 - 27x2 + 54x - 81, x0 = 3;
10)
f(x) = x5 - 6x4 + 13x3 - 14x2 + 12x - 8, x0 = 2;
11)
f(x) = x6 - 4x5 - 8x4 + 62x2 + 96x - 72, x0 = 2;
12)
f(x) = x5 + 4x4 + 7x3 + 7x2 + 4x + 1, x0 = -1;
13)
f(x) = 2x5 - 4x4 + x3 + x2 - 1, x0 = 1;
14)
f(x) = 3x6 - 19x5 + 33x4 + 23x3 - 130x2 - 132x - 40, x0 = 2;
15)
f(x) = x6 + 6x5 - 21x4 + 7x3 - x2 + 8 , x0 = 2;
16)
f(x) = x6 - 6x5 + 12x4 - 5x3 - 18x2 + 36x - 24, x0 = 2;
17)
f(x) = 3x6 - 5x5 + 3x4 - 7x3 + 2x2 + 10x - 6, x0 = 1;
18)
f(x) = -x6 + 6x5 - 12x4 + 5x3 + 18x2 - 36x + 24, x0 = 2;
19)
f(x) = x6 - 9x5 + 27x4 - 39x3 + 108x2 - 32x + 324, x0 = 3;
20)
f(x) = x6 - 5x5 + 5x4 + 6x3 + 4x2 - 40x + 32, x0 = 2;
Разложите многочлен f(x) по степеням двучлена g(x) = х - с, если:
1) пример решения
f(x) = 3х6 - 2x5 + 5x4 - 27x3 + 2x2 - 5, g(x) = x - 2;
2)
f(x) = 3х6 - 18x5 + 43x4 - 53x3 + 35x2 - 10х + 2, g(x) = x - 1;
3)
f(x) = х6 - ix5 + 2x4 + 3ix3 - x2 + 2ix + 1 + i, g(x) = x + i;
4)
f(x) = 2х6 - x5 + 2x3 - x2 + x + 1, g(x) = x - 2;
5)
f(x) = 2х6 - x5 + 3x4 - 4x3 + x2 - 2x + 1, g(x) = x - 3;
6)
f(x) = х6 + 2x5 - 12x4 + 30x3 - 80x - 10, g(x) = x - 3;
7)
f(x) = х6 - 7x5 - 8x4 + 450x2 + 2500x - 12000, g(x) = x - 5;
8)
f(x) = 2х7 - ix4 + 3x2 + (3 - i)x + 6, g(x) = x - 2i;
9)
f(x) = 2х5 - x4 + 3x2 - x + 6, g(x) = x - 1;
10)
f(x) = 2х5 + 7x4 - 8x2 + 3x - 5, g(x) = x + 2;
11)
f(x) = 3х6 - 2x5 + x4 + 2, g(x) = x + 1;
12)
f(x) = х5 - 9x4 + 28x3 - 36x2 + 27x - 27, g(x) = x - 3;
13)
f(x) = 2х5 - 7x3 + x2 + 2x - 5, g(x) = x + 2;
14)
f(x) = 2х6 - 7x5 + ix3 + (1 + i)x2 + 4x - 4i, g(x) = x + 2i;
15)
f(x) = х5 - 2x4 + 3x2 - 4x + 2, g(x) = x - 3i;
16)
f(x) = 4х6 - x5 + 2x4 - 3x2 + 6, g(x) = x + 4;
17)
f(x) = x5 + 2x4 - x3 + 2x2 - 1, g(x) = x - 2;
18)
f(x) = x5 + 4x4 + 7x3 + 7x2 + 4x + 1, g(x) = x - 1;
19)
f(x) = 2х5 - 9x4 + 29x3 - 56x2 + 63x - 30, g(x) = x - 3;
20)
f(x) = 2x5 + x4 - 2x3 - x2 + x + 2, g(x) = x - 2;