Find the multiplicity of a zero

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

Example #1

Find any multiple zeros of f(x) = x²(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) = x⁴ - 4x³ + 4x² and state the multiplicity of all zeros.

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

f(x) = x⁴ - 4x³ + 4x²

f(x) = x²(x² - 4x + 4)

f(x) = x²(x - 2)(x - 2)

f(x) = x²(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.