**Division Algorithm**

If and are polynomials, there exist unique polynomials and such that

whether (i) is the zero polynomial or (ii) deg < deg

**Remainder Theorem**

When a polynomial is divided by , the remainder is .

**Factor Theorem**

is a root of the equation if and only if is a factor of .

The Remainder and Factor Theorems are easy consequences of the Division Algorithm. By setting and applying the Division Algorithm, we see that is constant regardless of whether (i) or (ii) holds. Substituting into , we see that the constant value of is . It follows that if then is a factor of . Conversely, if is a factor of then .

**Problem**. Find the unique polynomial of degree three that satisfies and .

**Solution**. According to the Factor Theorem, has factors

, , and . It follows that there is a constant such that

To evaluate , we use the fact that . Thus

from which we find . Thus the desired polynomial is

If

then

and the sum of the roots is .