## The Factor and Remainder Theorems

Long division with polynomials sounds, and is, a great deal more complicated than long division with numbers. Fortunately though, it is not always necessary. There are two very helpful theorems which often turn the problem of long division into one of substitution.

The Factor Theorem:

If is a factor of then so is also a root of or equivalently, a solution of the equation The Factor Theorem is a special case of The Remainder Theorem.

The Remainder Theorem

The remainder when performing the long division of by is .If is a factor of then Example: Show that is a factor of  hence is a factor of Example. Find the remainder when is divided by We calculate Note is the solution to  More complicated questions may involve simultaneous equations: When is divided by the remainder is 4. When is divided by the remainder is 6. Find a and b. divided by remainder is 4  divided by remainder is 6 We now solve the simultaneous equations (1) (2)

3*(1)+(2) gives Then from (1)  