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.


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


Let  . Then   Q.E.D.


Let's divide   by  .

The theorem says that the remainder will be equal to  

So, let's divide   by  :  .

We get that  .


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.