Polynomial remainder theorem

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.

ProofEdit

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

 .

Let  . Then   Q.E.D.

ExampleEdit

Let's divide   by  .

The theorem says that the remainder will be equal to  

So, let's divide   by  :  .

We get that  .

UsesEdit

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.