數學一題...(急)

己知g(x) = px^3 - x^2 + qx + 6 可被x + 2 整除。當g(x) 除以x - 3時，餘式是30。

a)
g(-2) = p(-2)^3 - (-2)^2 + q(-2) + 6 = 0
g(3) = p(3)^3 - (3)^2 + q(3) + 6 = 30

p(-2)^3 - (-2)^2 + q(-2) + 6 = 0
-8p - 4 -2q + 6 = 0
-8p -2q + 2 = 0 -------(1)

p(3)^3 - (3)^2 + q(3) + 6 = 30
27p - 9 + 3q + 6 = 30
27p + 3q - 33 = 0
9p + q - 11 = 0 -------(2)

Solving (1) & (2), p = 2 & q = -7


b)
The equation become g(x) = 2x^3 - x^2 - 7x + 6 = 0

2x^3 - x^2 - 7x + 6 = (x + 2) (2x^2 - 5x + 3)
= (x-1)(x+2)(2x-3) = 0
x = -2 , 1, 3/2

good!