When we use long division to divide 1,573 by 12, we get a remainder of 1, so . A remainder of 1 means that 12 is not a factor of 1,573.

When we divide by 11 instead, we get a remainder of 0, so . A remainder of 0 means that 11 is a factor of 1,573.

The same thing happens with polynomials. While results in a remainder that is not 0, if we divide into , we do get a remainder of 0:

So is a factor of , while is not.

Earlier we learned that if is a factor of a polynomial , then , meaning is a zero of the function. It turns out that the converse is also true: If is a zero, then is a factor.

Now we know that if we start with a linear factor of a polynomial, then we know one of the zeros of the polynomial, and if we start with a zero of a polynomial, then we know one of the linear factors.

Lastly, even if is not a zero of , we can figure out what the remainder will be if we divide by , without having to do any division. If , then , so . So the remainder after division by is . This is the Remainder Theorem.