Question

Find the remainder of (h^4+h^2-2)/(h+3)

Answer 1

`(-3)^4+(-3)^2-2=88`, so by the polynomial remainder theorem, the remainder is 88.

To verify this, we can compute the quotient remainder:

`h^4=h^3\cdot h`, and
`h^3(h+3)=h^4+3h^3`. Subtracting from the numerator gives a remainder of
`-3h^3+h^2-2`

`-3h^3=-3h^2\cdot h`, and
`-3h^2(h+3)=-3h^3-9h^2`. Subtracting from the previous remainder gives a new remainder of
`10h^2-2`.

`10h^2=10h\cdot h`, and
`10h(h+3)=10h^2+30h`. Subtracting from the previous remainder gives a new remainder of
`-30h-2`.

`-30h=-30\cdot h`, and
`-30(h+3)=-30h-90`. Subtracting from the previous remainder gives a new remainder of
`88`, which doesn't contain factors of
`h`, so we're done.

This means we have

`(h^4+h^2-2)/(h+3)=h^3-3h^2+10h-30+(88)/(h+3)`