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:


`(p(m)+k)/(m+1)=(-2m^3+m^2-m-4)/(m+1)=-2m^2+3m-4`