# Zero theorem for the roots of a polynomial function

## The zero theorem lets you calculate the roots of a polynomial function

In this lesson we’ll learn how to use the zero theorem to calculate the roots of a factored polynomial.

We can use the zero theorem to find the roots of a polynomial function once it’s been factored. When a polynomial is factored, the zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???.

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For example, if you have the factored polynomial

???(2x+5)(x-3)=0???

then according to the zero theorem we can set each part equal to ???0??? to find any solutions to the equation.

We can say,

???2x+5=0???

???2x+5-5=0-5???

???2x=-5???

???\frac{1}{2}\cdot 2x = \frac{1}{2} \cdot -5???

???x=\frac{-5}{2}???

???x=-\frac{5}{2}???

and

???x-3=0???

???x-3+3=0+3???

???x=3???

## How to use the zero theorem to find the solutions for a polynomial function

Example

Find the solutions of the equation.

???y=x^2-13x+36???

The roots of the equation are where the ???y???-value equals ???0???.

We set up the equation

???x^2-13x+36=0???

and we’ll factor the left-hand side.

???(x-4)(x-9)=0???

Zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???. Therefore, we can say

???x-4=0???

???x-4+4=0+4???

???x=4???

and

???x-9=0???

???x-9+9=0+9???

???x=9???

The roots are ???x=4??? and ???x=9???.

We can use the zero theorem to find the roots of a polynomial function once it’s been factored.

## Another example of finding the roots of a function

Example

Find the zeros of the function.

???f(x)=5x^2-8x+3???

Finding the zeros of a function means finding the values of ???x??? when ???f(x)??? equals ???0???.

Let’s set the function equal to ???0??? and factor.

???5x^2-8x+3=0???

???(5x-3)(x-1)=0???

Zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???. Therefore, we can say

???5x-3=0???

???5x-3+3=0+3???

???5x=3???

???\frac{1}{5}\cdot 5x = \frac{1}{5} \cdot 3???

???x=\frac{3}{5}???

and

???x-1=0???

???x=1???

The zeros are ???x=3/5??? and ???x=1???.