Let f(x) = 3X^2 - 6X - 1 and g(x) = A(x-1)(x+1) + B(x - 1)^2 + C(x + 1)^2 .
When f(x) and g(x) are divided by x,x+1,and x+2 respectively,both f(x) and g(x) leave the same remainder
(a) find the vulues of A, B and C
(b) show that f(x) = g(x)
(c) slove g(x) = 8
FUNCTION
2007-01-09 12:03 am
回答 (2)
2007-01-09 12:19 am
✔ 最佳答案
(a) f(-1) = g(-1)3 + 6 - 1 = 4B
B = 2
f(0) = g(0)
-1 = -A + B + C
A - C = 2 + 1 = 3......(1)
f(-2) = g(-2)
12 + 12 - 1 = 3A + 9B + C
23 = 3A + 9(2) + C
3A + C = 5......(2)
(1) + (2): 4A = 8
A = 2
Sub. A = 2 into (1)
(2) - C = 3
C = -1
∴ A = 2, B = 2, C = -1
(b) RHS
= 2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2
= 2(x^2 - 1) + 2(x^2 - 2x + 1) - (x^2 + 2x + 1)
= 2x^2 - 2 + 2x^2 - 4x + 2 - x^2 - 2x - 1
= 3x^2 - 6x - 1
= LHS
∴ f(x) = g(x)
(c) g(x) = 8
2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2 = 8
3x^2 - 6x - 1 = 8
3x^2 - 6x - 9 = 0
x^2 - 2x - 3 = 0
(x + 1)(x - 3) = 0
x = -1 or x = 3
2007-01-09 12:30 am
(a) f(-1) = g(-1)
3 + 6 - 1 = 4B
B = 2
f(0) = g(0)
-1 = -A + B + C
A - C = 2 + 1 = 3......(1)
f(-2) = g(-2)
12 + 12 - 1 = 3A + 9B + C
23 = 3A + 9(2) + C
3A + C = 5......(2)
(1) + (2): 4A = 8
A = 2
Sub. A = 2 into (1)
(2) - C = 3
C = -1
∴ A = 2, B = 2, C = -1
(b) RHS
= 2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2
= 2(x^2 - 1) + 2(x^2 - 2x + 1) - (x^2 + 2x + 1)
= 2x^2 - 2 + 2x^2 - 4x + 2 - x^2 - 2x - 1
= 3x^2 - 6x - 1
= LHS
∴ f(x) = g(x)
(c) g(x) = 8
2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2 = 8
3x^2 - 6x - 1 = 8
3x^2 - 6x - 9 = 0
x^2 - 2x - 3 = 0
(x + 1)(x - 3) = 0
x = -1 or x = 3
3 + 6 - 1 = 4B
B = 2
f(0) = g(0)
-1 = -A + B + C
A - C = 2 + 1 = 3......(1)
f(-2) = g(-2)
12 + 12 - 1 = 3A + 9B + C
23 = 3A + 9(2) + C
3A + C = 5......(2)
(1) + (2): 4A = 8
A = 2
Sub. A = 2 into (1)
(2) - C = 3
C = -1
∴ A = 2, B = 2, C = -1
(b) RHS
= 2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2
= 2(x^2 - 1) + 2(x^2 - 2x + 1) - (x^2 + 2x + 1)
= 2x^2 - 2 + 2x^2 - 4x + 2 - x^2 - 2x - 1
= 3x^2 - 6x - 1
= LHS
∴ f(x) = g(x)
(c) g(x) = 8
2(x - 1)(x + 1) + 2(x - 1)^2 - (x + 1)^2 = 8
3x^2 - 6x - 1 = 8
3x^2 - 6x - 9 = 0
x^2 - 2x - 3 = 0
(x + 1)(x - 3) = 0
x = -1 or x = 3
收錄日期: 2021-04-12 21:26:08
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