有關因式定理既問題(20點)

1a)
當f(x)除以2x+1時,餘數為5/4
=>f(-1/2)=5/4
=>6(-1/2)^3-p(-1/2)^2+q(-1/2)+1=5/4
=>-p/4-q/2=1
=>-p-2q=4
=>p=-2q-4...(1)

當f(x)除以3x-2時,餘數為-1/9
=>f(2/3)=-1/9
=>6(2/3)^3-p(2/3)^2+q(2/3)+1=-1/9
=>-4p/9+2q/3=-26/9
=>-4p+6q=-26...(2)

代(1)入(2):
-4(-2q-4)+6q=-26
8q+16+6q=-26
14q=-42
q=-3
p=-2(-3)-4=2

b)
當f(x)除以x+2所得的餘數
=f(-2)
=6(-2)^3-2(-2)^2-3(-2)+1
=-49

2)
2x^3+ax^2+bx-90
=(2x^3+5x^2-18x)+[(a-5)x^2+(b+18)x-90]
=x(2x^2+5x-18)+5[(a-5)x^2/5+(b+18)x/5-18]

由於2x^3+ax^2+bx-90可被2x^2+5x-18整除,所以:
(a-5)x^2/5+(b+18)x/5-18=2x^2+5x-18

(a-5)/5=2 => a-5=10 => a=15
(b+18)/5=5 => b+18=25 => b=7