What is the remainder when is divided by ?

Source: NCTM Mathematics Teacher, November 2006

**SOLUTION**

If a polynomial is divided by a polynomial , there exists a quotient polynomial and a remainder polynomial such that

with .

Though tedious it is not hard to do the long division by . A first few steps reveal that the quotient equals . Since we are after the remainder and not the quotient, let’s focus our attention on the last stage of the operation. It looks like the following

The remainder equals .

**Answer**:

**Alternative Solution**

The divisor . We are going to make two divisions, first by then by . We will get two remainders from which we calculate the final remainder.

The first division (either long or synthetic) by yields a remainder of

The second division of the above expression by yields a remainder of .

The rational expressions of the two remainders are and .

The remainder equals .