Factor theorem

In algebra, the factor theorem is a theorem for finding out the factors of a polynomial (an expression in which the terms are only added, subtracted or multiplied, e.g. x² + 6x + 6). It is a special case of the polynomial remainder theorem.

The factor theorem states that a polynomial f(x) has a factor x − k if and only if f(k) = 0.

An example

You wish to find the factors of

x³ + 7x² + 8x + 2.

To do this you would use trial and error finding the first factor. When the result is equal to 0, we know that we have a factor. Is (x − 1) a factor? To find out, substitute x = 1 into the polynomial above:

x³ + 7x² + 8x + 2 = (1)³ + 7(1)² + 8(1) + 2

= 1 + 7 + 8 + 2

= 18

As this is equal to 18—not 0—(x − 1) is not a factor of x³ + 7x² + 8x + 2. So, we next try (x + 1) (substituting x = − 1 into the polynomial):

( − 1)³ + 7( − 1)² + 8( − 1) + 2.

This is equal to 0. Therefore x − ( − 1), which is to say x + 1, is a factor, and -1 is a root of x³ + 7x² + 8x + 2.

The next two roots can be found by algebraically dividing x³ + 7x² + 8x + 2 by (x + 1) to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic equation. = x² + 6x + 2 and therefore (x + 1) and x² + 6x + 2 are the factors of x³ + 7x² + 8x + 2.