Question

The smallest integer that can be added to -2m3 − m + m2 + 1 to make it completely divisible by m + 1 is .

Answer 1

Recall the polynomial remainder theorem: if $m-c$ is a factor of some polynomial $p(m)$, then the remainder upon dividing $p(m)$ by $m-c$ is $p(c)$.

Let $c=-1$ and $p(m)=-2m^3+m^2-m+1$. The remainder left from dividing $p(m)$ by $m+1$ is

$p(-1)=-2(-1)^3+(-1)^2-(-1)+1=5$

If we subtract 5 from both sides, we'd get a "remainder" of 0, which suggests that we have to add -5 to $p(m)$ to make $m+1$ a factor.