f.4 (polynomial)

f(x) = (x-1)(3x+2)(a+bx) + (px+q)
If f(x) is divided by x-1,the remainder is -4.
then,
f(1) = -4
(1-1)(3+2)(a+b) + (p+q) = -4
p + q = -4 ..............................(1)

If f(x) is divided by 3x+2,the remainder is 6.
then,
f(-2/3) = 6
(-2/3 - 1)(3(-2/3) + 2)(a + b(-2/3)) + (p(-2/3) + q = 6
-2p/3 + q = 6 ...........................(2)

(1) - (2)
p + 2p/3 = -4 -6
5p/3 = -10
p = -10 x 3 / 5
p = -6

Sub p= -6 into (1)
-6 + q = -4
q = -4 - (-6)
q = -4 + 6
q =2

When f(x) is divided by x-1,the remainder is -4.
then f(1)=-4 by (餘式定理)
又因為f(x)=(x-1)(3x+2)g(x)+px+q by (因式定理)
所以f(1)=p+q=-4
When f(x) is divided by 3x+2,the remainder is 6.
then f(-2/3)=6
又因為f(x)=(x-1)(3x+2)g(x)+px+q
所以f(-2/3)=-2/3p+q=6
slove : p+q=-4 and -2/3p+q=6
slove that p=-6, q=2