What is the remainder when x^{2013} is divided by x^{2}– 1? This problem was taken from 2009 MTAP-Dep-Ed Math Challenge Finals. The problem looks very easy but if you don’t know the basic principle how to deal with this problem you can never answer this in just a minute. This article will teach you how to deal with problem like this.

be a polynomial with degree n.

be the divisor in second degree.

be the quotient when is divided by

be the remainder when is divided by

If is a 1^{st} degree expression, the remainder is constant.

If is a second degree expression and the remainder is a in 1^{st} degree expression in the form of .

We can say that

Worked Problem 1:

What is the remainder when is divided by

Given:

Required:

Solution:

Let to simplify things out.

, let’s call this equation 1.

Let

let’s call this equation 2.

Solve equation 1 and 2 simultaneously.

Therefore the remainder is or .

Worked Problems 2:

What is the remainder when is divided by ?

Solution:

Given:

Let x=2

, equation 1

Let x=0

From equation 1,

,

The remainder is or simply