# Using the remainder theorem to find a remainder.

## Question

The polynomial  is divided by . What is the remainder?

## Solution

Do you want to try this problem using polynomial long division? That could be a nightmare. How about synthetic division? I didn’t think so. You need the remainder theorem.

The remainder theorem states that when a polynomial (like, for example, the one in this problem) is divided by a binomial , the remainder is equal to the value of the polynomial at point .

Consider a simple example: When polynomial  is divided by , the remainder is equal to the value of  at point . Since 1 is an easy number at which to evaluate a function, it’s not too hard to see that at ,   Thus when  is divided by , the remainder is

You can confirm this result with polynomial long division or synthetic division.

Now try the question. To find the remainder when  is divided by , evaluate   at  That’s

Note that -1 taken to an even power is 1 and -1 taken to an even power is 1. Thus , , and ; while  and . The whole polynomial equals

The remainder is 1.

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