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.