Question

When kx^3+px^2-x+3 is divided by x-1, the remainder is 4. When kx^3+px^2-x+3 is divided by x-2, the remainder is 21. Find the values of k and p.

Answer 1

Use the polynomial remainder theorem.


`(kx^3+px^2-x+3)/(x-1)=q(x)+\frac4{x-1}\implies kx^3+px^2-x+3=q(x)(x-1)+4`

`x=1\implies k+p-1+3=0+4\implies k+p=2`


`(kx^2+px^2-x+3)/(x-2)=q(x)+(21)/(x-2)\implies kx^3+px^2-x+3=q(x)(x-2)+21`

`x=2\implies 8k+4p-2+3=0+21\implies 2k+p=5`

Now solve the system


`\begin{cases}k+p=2\\2k+p=5\end{cases}\implies k=3,p=-1`