Concept

Multiplicity of Zeros

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} - 5 x^{2} + 7 x - 3

This function can be rewritten as follows.

p (x) = x^{3} - 5 x^{2} + 7 x - 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.

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