This lesson will show how to find the multiplicity of a zero with a few good examples.

The zero of a polynomial function is the value or number that will make the polynomial function equal to 0. When the zero of a polynomial function is repeated, it is called a **multiple zero**. The **multiplicity of a zero** is equal to the number of times the zero is repeated.

**Example #1**

Find any multiple zeros of f(x) = x^{2}(x + 10)(x - 5)^{3} and state the multiplicity of all zeros.

The zeros of the function are 0, -10, and 5 since these values will make the function equal to 0. The numbers 0 and 5 are multiple zeros of the polynomial function.

The multiplicity of 0 is 2 and the multiplicity of 5 is 3

**Example #2**

Find any multiple zeros of f(x) = x(x + 1)^{5}(x - 3)^{4}(2x + 4)^{6} and state the multiplicity of all zeros.

The zeros of the function are 0, -1, 3 and -2 since these values will make the function equal to 0. The numbers -1, 3, and -2 are multiple zeros of the polynomial function.

The multiplicity of -1 is 5, the multiplicity of 3 is 4, and the multiplicity of -2 is 6

**Example #3**

Find any multiple zeros of f(x) = x^{4} - 4x^{3} + 4x^{2} and state the multiplicity of all zeros.

First, factor the function so you can clearly identify the zeros.

f(x) = x^{4} - 4x^{3} + 4x^{2}

f(x) = x^{2}(x^{2} - 4x + 4)

f(x) = x^{2}(x - 2)(x - 2)

f(x) = x^{2}(x - 2)^{2}

The zeros of the function are 0 and 2 since these values will make the function equal to 0. The numbers 0 and 2 are multiple zeros of the polynomial function.

The multiplicity of 0 is 2 and the multiplicity of 2 is 2.