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

QUESTION POSTED AT 01/06/2020 - 03:24 PM

Answered by admin AT 01/06/2020 - 03:24 PM

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
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