Polynomial remainder theorem

If we divided ${\displaystyle P(x)}$ by ${\displaystyle x-a}$, it means that we found such ${\displaystyle Q(x)}$ and ${\displaystyle b}$, that

${\displaystyle P(x)=Q(x)\cdot (x-a)+b}$.

Let ${\displaystyle x=a}$. Then ${\displaystyle P(a)=Q(a)\cdot (a-a)+b=0+b=b.}$ Q.E.D.

Let's divide ${\displaystyle P(x)=x^{2}-2x+4}$ by ${\displaystyle x-6}$.

The theorem says that the remainder will be equal to ${\displaystyle (6)^{2}-2\cdot (6)+4=36-12+4=28.}$

So, let's divide ${\displaystyle P(x)-28=x^{2}-2x-24}$ by ${\displaystyle x-6}$: ${\displaystyle x^{2}-2x-24=(x-6)(x+4)}$.

We get that ${\displaystyle x^{2}-2x+4=(x+4)(x-6)+28}$.

Sometimes, it can be difficult to calculate ${\displaystyle P(a)}$ fast (if the polynomial is very big). There exist methods of fast division by ${\displaystyle x-a}$, so the computing the remainder could be easier than the whole function.

The theorem itself is also very important for theoretical use. Its proof can easily be applied to another similar objects with little changes. It is used, for example, in the fundamental theorem of algebra, in the form of a generalisation in complex numbers.