Learn how to find zeros of a polynomial function with a few well-chosen examples.

**Example #1**

Find the zeros of f(x) = (x - 4)(x + 2)(x + 5)

Using the zero product property, find a zero for each linear factor

x - 4 = 0

x - 4 + 4 = 0 + 4

**x = 4**

x + 2 = 0

x + 2 - 2 = 0 - 2

**x = -2**

x + 5 = 0

x + 5 - 5 = 0 - 5

**x = -5**

The zeros of the function f(x) = (x - 4)(x + 2)(x + 5) are 4, -2, and -5

**Example #2**

Find the zeros of f(x) = x^{4} - x^{2}

We cannot use the zero product property yet since we cannot see the linear factors.

Rewrite the function f(x) = x^{4} - x^{2} to show its linear factors. We can do this by factoring the function completely.

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

f(x) = x^{2}(x^{2} - 1)

f(x) = (x)(x)(x - 1)(x + 1)

Now you can use the zero product property to find the zero for each linear factor.

x = 0

x = 0

x - 1 = 0

x - 1 + 1 = 0 + 1

**x = 1**

x + 1 = 0

x + 1 - 1 = 0 - 1

**x = -1**

The zeros of the function f(x) = x^{4} - x^{2 }are 0, 1, and -1

**Example #3**

Find the zeros of f(x) = 8x^{3} - 8x^{2} - 160x

We cannot use the zero product property yet since we cannot see the linear factors.

Rewrite the function f(x) = 8x^{3} - 8x^{2} - 160x to show its linear factors. We can do this by factoring the function completely.

f(x) = 8x^{3} - 8x^{2} - 160x

f(x) = 8x(x^{2} - x - 20)

f(x) = 8x(x - 5)(x + 4)

Using the zero product property, find a zero for each linear factor

8x = 0

**x = 0**

x - 5 = 0

x - 5 + 5 = 0 + 5

**x = 5**

x + 4 = 0

x + 4 - 4 = 0 - 4

**x = -4**

The zeros of the function f(x) = 8x^{3} - 8x^{2} - 160x are 0, 5, and -4