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F.4餘式定理,因式定理

發問:

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

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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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