If *P*( *x*) is a polynomial, then *P*( *r*) = 0 if and only if *x* – *r* is a factor of *P*( *x*).

##### Example 1

Is ( *x* + 2) a factor of *x* ^{3} – *x* ^{2} – 10 *x* – 8?

Check to see whether ( *x* ^{3} – *x* ^{2} – 10 *x* – 8) ÷ ( *x* + 2) has a remainder of zero. Using synthetic division, you get

Because the remainder of the division is zero, ( *x* + 2) is a factor of *x* ^{3} – *x* ^{2} – 10 *x* – 8. The expression *x* ^{3} – *x* ^{2} – 10 *x* – 8 can now be expressed in factored form.

*x* ^{3} – *x* ^{2} – 10 *x* – 8 = ( *x* + 2)( *x* ^{2} – 3 *x* – 4)

But ( *x* ^{2} – 3 *x* – 4) can be factored further into ( *x* – 4)( *x* + 1). So

*x* ^{3} – *x* ^{2} – 10 *x* – 8 = ( *x* + 2)( *x* – 4)( *x* + 1)

The expression *x* ^{3} – *x* ^{2} – 10 *x* – 8 is now **completely factored.** From this form, it also is seen that ( *x* – 4) and ( *x* + 1) are also factors of *x* ^{3} – *x* ^{2} – 10 *x* – 8.