1.若3x=A(x-1)(x+2)+B(x-2)(x+2)+C(x+1)(x-1),求A,B和C的值
2.當px^3+qx^2-14x-37除以4x^2-5時,商式是3x+8,餘式是x+3。求p和q的值
3.若f(x)=2x^-2x-3除以x-q的餘式是q^2,求q的值
F.4餘式定理,因式定理
2010-02-26 7:59 pm
回答 (1)
2010-02-26 8:15 pm
✔ 最佳答案
Q1. 3x = (A + B + C)x^2 + Ax + (-2A - 4B - C)so A = 3.
3 + B + C = 0 ......(1)
-6 - 4B - C = 0 .......(2)
(1) + (2) we get - 3 - 3B = 0, so B = -1
from (1), 3 - 1 + C = 0, so C = - 2.
Q2.
px^3 + qx^2 - 14x - 37 = (4x^2 - 5)(3x + 8) + (x + 3) = 12x^3 + 32x^2 + ...
so p = 12 and q = 32.
Q3.
f(x) = 2x^2 - 2x - 3 = (x - q)Q(x) + q^2.
Put x = q, we get
2q^2 - 2q - 3 = (q - q)Q(q) + q^2
2q^2 - 2q - 3 = q^2
q^2 - 2q - 3 = 0
(q - 3)(q + 1) = 0
so q = 3 or - 1.
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