## Finding the zeros of a polynomial from a graph

The zeros of a polynomial are the solutions to the equation p(x) = 0, where p(x) represents the polynomial. If we graph this polynomial as y = p(x), then you can see that these are the values of x where y = 0. In other words, they are the x-intercepts of the graph.

The zeros of a polynomial can be found by finding where the graph of the polynomial crosses or touches the x-axis.

Let’s try this out with an example!

## Example

Consider a polynomial f(x), which is graphed below. What are the zeros of this polynomial?

To answer this question, you want to find the x-intercepts. To find these, look for where the graph passes through the x-axis (the horizontal axis).

This shows that the zeros of the polynomial are: x = –4, 0, 3, and 7.

While here, all the zeros were represented by the graph actually crossing through the x-axis, this will not always be the case. Consider the following example to see how that may work.

## Example

Find the zeros of the polynomial graphed below.

As before, we are looking for x-intercepts. But, these are any values where y = 0, and so it is possible that the graph just touches the x-axis at an x-intercept. That’s the case here!

From here we can see that the function has exactly one zero: x = –1.

## Connection to factors

You may remember that solving an equation like f(x) = (x – 5)(x + 1) = 0 would result in the answers x = 5 and x = –1. This is an algebraic way to find the zeros of the function f(x). Each of the zeros correspond with a factor: x = 5 corresponds to the factor (x – 5) and x = –1 corresponds to the factor (x + 1).

So if we go back to the very first example polynomial, the zeros were: x = –4, 0, 3, 7. This tells us that we have the following factors:

(x + 4), x, (x – 3), (x –7)

However, without more analysis, we can’t say much more than that. For example, both of the following functions would have these factors:

f(x) = 2x(x+4)(x–3)(x–7)

and

g(x) = x(x+4)(x–3)(x–7)

In the second example, the only zero was x = –1. So, just from the zeros, we know that (x + 1) is a factor. If you have studied a lot of algebra, you recognize that the graph is a parabola and that it has the form $a(x+1)^2$, where a > 0. But only knowing the zero wouldn’t give you that information*.

*you can actually tell from the graph AND the zero though. Since the graph doesn’t cross through the x-axis (only touches it), you can determine that the power on the factor is even. But, this is a little beyond what we are trying to learn in this guide!