Find the remainder of a high degree polynomial

Question

If $$f(x)=(x-1)^{2017}+(x-3)^{2016}+x^2+x+1$$ and $$g=x^2-4x+4$$ find the remainder of f divided by g. I only found that $$g=(x-2)^2$$ but I don't know how to go further. If I set $$x=2$$ then $$f(2)=9$$ How to use this? Typo:$$ f(2)=9$$

Answer 1

We have $f(x)=(x-2)^2P(x)+ ax+b$, and we wish to find $a, b$. As you’ve already found, $f(2)=9$, so we also have $2a+b=9$.

The trick here is to differentiate $f(x)$ to obtain $f’(x) = 2(x-2)P(x) + (x-2)^2P’(x) + a$. Substituting $x=2$ gives $a=f’(2)$. Computing this, we obtain $a=6$. Thus $b=-3$ and we’re done.