form4數學題一問

Let f(x) = x^3 + px^2 + qx + 2 and
g(x) = x^3 + qx + px - 6

Since (x - 2) is the common factor of both f(x) and g(x)
Hence, f(2) = g(2) = 0

(2)^3 + p(2)^2 + q(2) + 2 = 0
8 + 4p + 2q + 2 = 0
4p + 2q + 10 = 0
2p + q + 5 = 0 ----------------(1)

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

(1) - (2)
p + 4 = 0
p = -4

From (2)
-4 + q + 1 = 0
q - 3 = 0
q = 3