Question

If ` f(x) = x^4 - 2 x^3 + 3 x^2 -ax +b` is a polynomial such that when it is divided by x-1 and x+1 the remainders are respectively 5 and 19. Determine the remainder when f(x) is divided by x-2.

Answer 1

`f(x)=x^4-2x^3+3x^2-ax+b`
`f(1)=1-2(1)+3(1)-a+b`
`b-a=3-(1)`
`f(-1)=1-2(-1)+3(1)-a(-1)+b` `19=1+2+3+a+b`
`a+b=13-(2)`
adding equation 1 and 2
`2b=16`
`b=8`
`a=8-3=5`
f(2)=16-2*8+3*4-5*2+8
=16-16+12-10+8=10.