Using the remainder theorem to find a remainder.
The polynomial is divided by . What is the remainder?
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.