Everyone knows the joy of finding solutions to simple and complex polynomials. It seems easy in the beginning. But as the polynomials become more complex, finding solutions becomes just as complicated. But it doesn’t have to be.
The rational root theorem states that when trying to find solutions or zeros for a polynomial, the potential solutions will be composed of a factor of the constant, divided by a factor of the highest power’s coefficient. All solutions will take the form of a rational number.
The rational root theorem is one way to break through the mud so that you can solve polynomials with ease. Let’s break the rational root theorem down a bit to understand it better.
Deeper dive into the rational root theorem.
As you progress in your math class, the polynomials will become more complex. Sometimes, when we are asked to solve a complex polynomial, we get tunnel vision on what the possible solutions are. Using the rational root theorem opens up the world of possibilities. This tool is used from Algebra up to higher mathematics.
What are rational roots, and what are rational numbers? Let’s start with rational numbers.
Rational numbers consist of a whole integer over a whole integer. A rational number is a number written as a fraction that will either terminate or will have a repeating pattern. In contrast, an irrational number does not have that repeating pattern and does not terminate. Here are a few examples.
A rational root of a polynomial is the value you get when solving x. When you are asked to find the x-intercepts of a polynomial when graphing, you are setting f(x) =0 and solving for x. The x values are the rational roots that we are calculating.
Here are just a few examples of how the rational root theorem can be helpful.
- It can be helpful when asked to find the zeros of a complex polynomial.
- It can be used to help locate intercepts when graphing a polynomial.
The main idea is to find the zeros or roots of a polynomial.
The theory states that the potential roots take on the following form.
We have to make a few things clear here. The value of an will always be the coefficient of the variable with the highest power in our polynomial. The value of a0 will always be the constant.
Not all polynomials will have rational roots, and that’s ok. Sometimes polynomials cannot be solved, or they have no rational solutions.
How the rational root theorem works.
We are going to start off by going through a problem step-by-step. That way, you can see how we develop all the possible rational roots for a problem.
The polynomial we will work with is 3x5+2x4-3x3-2x2+x-4.
First, we locate our constant, which is -4. We have to list all the factors, both positive and negative, that make up -4. Our list includes the values +1, -1, +2, -2, +4, and -4.
Then we locate the coefficient of the highest-powered variable in f(x). This would be 3. So we list all of the factors that make up 3. Our list includes the values +1, -1, +3, and -3.
All of our possible rational solutions or roots will take on the form of:
Once we have completed this list, we can knock out or delete any duplicates from our list.
We can then take our list and test to see which ones zero out our polynomial.
Example problems using the rational root theorem.
Here are a few problems worked out that will hopefully make this theorem a bit easier to understand.
In the first example, we are going to find the roots of the polynomial, f(x) = x6 – 2x4 + 3x2 + 2x -8.
Our first step is to find the factors of the coefficient of the highest power. The factors of 1 are ±1.
The next step is to find the factors of the constant. The constant is -8. The factors of -8 are ±1, ±2, ±4, and ±8.
Our potential factors take on the form: factor of 8 / factor of 1. So our potential factors roots for f(x) are ±1, ±2, ±4, and ±8.
For our second example problem, we will work with the polynomial f(x) = 3x5+2x4-3x3-2x2+x-4.
Our first step is to find the factors of the coefficient of the highest power. The factors of 8 are ±1, ±2, ±4, and ±8.
The next step is to find the factors of the constant. The constant is 12. The factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12.
Our potential factors take on the form: factor of 8 / factor of 1. So our potential factors roots for f(x) are ±1, ± 2, ± 3, ± 4, ± 6, ± 12, ± 1/2, ± 3/2, ± 1/4, ± 1/8, ± 3/4, and ± 3/8.
Once you find your potential roots, you can see if they are actual zeros of the polynomial. You can go about this two ways.
You can plug the values back into the polynomial to see if the polynomial calculates to zero.
Or you can use take the potential values and use Synthetic Division to determine if they actually are root factors of the polynomial. Either way will work.
Remember that sometimes there will be polynomials with no rational roots, and that’s ok.
Keep practicing. And don’t give up!