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The Factor Theorem

Generally when a polynomial is divided by a binomial there is a remainder.

Consider the polynomial function f (x) = x³ + 6 x² – x – 30. Divide the polynomial f (x) by the binomial x + 3.

Observe that, the remainder is 0.

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 x² + 3 x – 10.

From the results of the division and by using the Remainder Theorem, we can write the following statement.

x³ + 6 x²– x – 30 = ( x² + 3 x – 10)( x + 3) + 0.

Since the remainder is 0, the function value at –3 is 0 or f (–3) = 0. This means that the binomial x + 3 is a factor of the polynomial function f (x) = x³ + 6 x²– x – 30.

This illustrates the Factor Theorem.

A polynomial f (x) has a factor (x – k) if and only if f (k) = 0 where f (x) is a polynomial of degree and k is any real number.