The **zero of a function** is any replacement for the variable that will produce an answer of zero. Graphically, the real zero of a function is where the graph of the function crosses the *x*‐axis; that is, the real zero of a function is the *x*‐intercept(s) of the graph of the function.

##### Example 1

Find the zeros of the function *f* ( *x*) = *x* ^{2} – 8 *x* – 9.

Find *x* so that *f* ( *x*) = *x* ^{2} – 8 *x* – 9 = 0. *f* ( *x*) can be factored, so begin there.

Therefore, the zeros of the function *f* ( *x*) = *x* ^{2} – 8 *x* – 9 are –1 and 9. This means

*f* (–1) = 0 and *f* (9) = 0

If a polynomial function with integer coefficients has real zeros, then they are either rational or irrational values. Rational zeros can be found by using the rational zero theorem.