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 $0$.

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

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

$x-3=0$

$x-3+3=0+3$

$x=3$

Step-by-step examples using the zero theorem

---

Take the course

Want to learn more about Algebra 2? I have a step-by-step course for that. :)

---

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 $x=1$.

---

Get access to the complete Algebra 2 course

---