Question

What is the smallest integer that can be added to -2m^3-m+m^2+1 to make it completely divisible by m+1?

Answer 1

Let $p(m)=-2m^3+m^2-m+1$. We want to find the least $k\in\mathbb Z$ such that $p(m)+k$ has remainder 0 when divided by $m+1$.

By the polynomial remainder theorem, this will happen if $p(m)+k=0$ when $m=-1$:

$p(-1)+k=-2(-1)^3+(-1)^2-(-1)+1+k=0\implies k=-5$

We can check this:

$\dfrac{p(m)+k}{m+1}=\dfrac{-2m^3+m^2-m-4}{m+1}=-2m^2+3m-4$