If and are polynomials, there exist unique polynomials and such that
whether (i) is the zero polynomial or (ii) deg < deg
When a polynomial is divided by , the remainder is .
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 .
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
and the sum of the roots is .