Remainder Theorem With Second Degree Divisor
What is the remainder when x2013 is divided by x2– 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 1st degree expression, the remainder is constant.
If is a second degree expression and the remainder is a in 1st degree expression in the form of .
We can say that
Worked Problem 1:
What is the remainder when is divided by
Let to simplify things out.
, let’s call this equation 1.
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 ?
, equation 1
From equation 1,
The remainder is or simply