The Factor Theorem
Generally when a polynomial is divided by a binomial there is a remainder.
Consider the polynomial function . Divide the polynomial by the binomial .
Observe that, the remainder is .
When you divide a polynomial by one of its binomial factors, the quotient is called a depressed polynomial.
Here the quotient or the depressed polynomial is .
From the results of the division and by using the Remainder Theorem , we can write the following statement.
Since the remainder is , the function value at is or . This means that the binomial is a factor of the polynomial function .
This illustrates the Factor Theorem.
A polynomial has a factor if and only if where is a polynomial of degree and is any real number.