Polynomial remainder theorem

The polynomial remainder theorem states that:

for every polynomial  and every real number  the remainder of division of  by  is .

This theorem can easily be proven, but it is important for various calculations.

Proof
If we divided  by , it means that we found such  and , that

.

Let . Then  Q.E.D.

Example

Let's divide  by .

The theorem says that the remainder will be equal to 

So, let's divide  by : .

We get that .

Uses
Sometimes, it can be difficult to calculate  fast (if the polynomial is very big). There exist methods of fast division by , 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.

Category:Mathematical theorems
Category:Algebra