How To Find The Roots Of A Polynomial

The roots of a polynomial are also called its zeroes, because the roots are the ​x​ values at which the function equals zero. When it comes to actually finding the roots, you have multiple techniques at your disposal; factoring is the method you'll use most frequently, although graphing can be useful as well.

Examine the highest-degree term of the polynomial – that is, the term with the highest exponent. That exponent is how many roots the polynomial will have. So if the highest exponent in your polynomial is 2, it'll have two roots; if the highest exponent is 3, it'll have three roots; and so on.

The most versatile way of finding roots is factoring your polynomial as much as possible, and then setting each term equal to zero. This makes a lot more sense once you've followed through a few examples. Consider the simple polynomial ​x​² – 4​x:​

1. Factor the Polynomial

A brief examination shows that you can factor ​x​ out of both terms of the polynomial, which gives you:

\(x(x – 4)\)

2. Find the Zeroes

Set each term to zero. That means solving for two equations:

\(x = 0\)

is the first term set to zero, and

\(x – 4 = 0\)

is the second term set to zero.

You already have the solution to the first term. If ​x​ = 0, then the entire expression equals zero. So ​x​ = 0 is one of the roots, or zeroes, of the polynomial.

Now, consider the second term and solve for ​x​. If you add 4 to both sides you'll have:

\(x – 4 + 4 = 0 + 4\)

which simplifies to:

\(x = 4\)

So if ​x​ = 4 then the second factor is equal to zero, which means the entire polynomial equals zero too.

3. List Your Answers

Because the original polynomial was of the second degree (the highest exponent was two), you know there are only two possible roots for this polynomial. You've already found them both, so all you have to do is list them:

\(x = 0, x = 4\)

Here's one more example of how to find roots by factoring, using some fancy algebra along the way. Consider the polynomial ​x​⁴ – 16. A quick look at its exponents shows you that there should be four roots for this polynomial; now it's time to find them.

1. Factor the Polynomial

Did you notice that this polynomial can be rewritten as the difference of squares? So instead of ​x​⁴ – 16, you have:

\((x^2)^2 – 4^2\)

Which, using the formula for the difference of squares, factors out to the following:

\((x^2 – 4)(x^2 + 4)\)

The first term is, again, a difference of squares. So although you can't factor the term on the right any further, you can factor the term on the left one step more:

\((x – 2)(x + 2)(x^2 + 4)\)

2. Find the Zeroes

Now it's time to find the zeroes. It quickly becomes clear that if ​x​ = 2, the first factor will equal zero, and thus the entire expression will equal zero.

Similarly, if ​x​ = −2, the second factor will equal zero and thus so will the entire expression.

So ​x​ = 2 and ​x​ = −2 are both zeroes, or roots, of this polynomial.

But what about that last term? Because it has a "2" exponent, it should have two roots. But you can't factor this expression using the real numbers you're used to. You'd have to use a very advanced mathematical concept called imaginary numbers or, if you prefer, complex numbers. That's far beyond the scope of your current math practice, so for now it's enough to note that you have two real roots (2 and −2), and two imaginary roots that you'll leave undefined.

You can also find, or at least estimate, roots by graphing. Every root represents a spot where the graph of the function crosses the ​x​ axis. So if you graph out the line and then note the ​x​ coordinates where the line crosses the ​x​ axis, you can insert the estimated ​x​ values of those points into your equation and check to see if you've gotten them correct.

Consider the first example you worked, for the polynomial ​x​² – 4​x​. If you draw it out carefully, you'll see that the line crosses the ​x​ axis at ​x​ = 0 and ​x​ = 4. If you input each of these values into the original equation, you'll get:

\(0^2 – 4(0) = 0\)

so ​x​ = 0 was a valid zero or root for this polynomial.

\(4^2 – 4(4) = 0\)

so ​x​ = 4 is also a valid zero or root for this polynomial. And because the polynomial was of degree 2, you know you can stop looking after finding two roots.