Dividing Polynomials
Rule

The Factor Theorem

Let be a polynomial and a real number. The binomial is a factor of if and only if
The binomial is a factor of if and only if

Notice that means that is a zero of Therefore, the theorem can also be stated as follows.

The binomial is a factor of if and only if is a zero of

The Factor Theorem is a special case of the Remainder Theorem and establishes a connection between the zeros of a polynomial and its factors.

Proof

Since the theorem is a biconditional statement, the proof will consists of two parts.

  • Part I: If is a factor of then
  • Part II: If then is a factor of

Part I

If is a factor of then can be written as the product of and a certain polynomial
Now, evaluate the equation at
Consequently, is a zero of This completes the proof of the first part.

Part II

Consider the division of and
The division can be rewritten in terms of the quotient and the remainder by using polynomial long division.
By the Remainder Theorem, the remainder of the previous division is equal to Since the remainder of the division is Therefore, the rightmost term of the previous equation is
Finally, multiply both sides of the equation by
Consequently, is a factor of This completes the proof of the second part.