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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) = x4 - x2
We cannot use the zero product property yet since we cannot see the linear factors.
Rewrite the function f(x) = x4 - x2 to show its linear factors. We can do this by factoring the function completely.
f(x) = x4 - x2
f(x) = x2(x2 - 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) = x4 - x2 are 0, 1, and -1
Example #3
Find the zeros of f(x) = 8x3 - 8x2 - 160x
We cannot use the zero product property yet since we cannot see the linear factors.
Rewrite the function f(x) = 8x3 - 8x2 - 160x to show its linear factors. We can do this by factoring the function completely.
f(x) = 8x3 - 8x2 - 160x
f(x) = 8x(x2 - 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) = 8x3 - 8x2 - 160x are 0, 5, and -4