In polynomial functions, the multiplicity of a zero is the number of times that a zero can be factored from a function. Using the Factor Theorem, it is possible to rewrite a polynomial as a multiplication of its factors. As an example, consider the following polynomial function. p(x) = x^3-5x^2+7x-3 This function can be rewritten as follows. p(x) = x^3-5x^2+7x-3 ⇓ p(x) = (x-3)(x-1)(x-1) This means that the zeros of p(x) are 1 and 3. In this case, even though there are three factors, there are only two unique zeros. Since 1 can be factored twice in the polynomial, the zero x=1 has a multiplicity of 2. Consider the graph of the polynomial function.

The graph of a polynomial function crosses the x-axis on zeros with an odd multiplicity. If the multiplicity of a zero is even, the graph only touches the x-axis.