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.