The Remainder Theorem

You can find a remainder without dividing. ${x}^{2}+5x+9$ divided by $x+2$ has remainder 3. Now put x = -2 into ${x}^{2}+5x+9$: $4-10+9 = 3$. It is 3 again.

This always works. Any division can be written as $p\left(x\right) = \left(x-a\right)q\left(x\right)+r$, where r is the remainder. Put x = a. The bracket is 0, so the product is 0. Only r is left.

This is the remainder theorem: dividing by $x-a$ leaves the remainder $p\left(a\right)$. For $x+2$, the remainder is $p\left(-2\right)$.