Rational zeroes theorem
Rational Zero Theorem
What is the Rational Zero Theorem:
When trying to find the roots or solutions to a polynomial, there are several methods one can use. Commonly used methods include factoring and the quadratic formula. Before we get into the rational zero theorem, otherwise known as the rational root theorem, it is important you understand other method of solving polynomials, including solving quadratic equations by completing the square and solving quadratic equations using the quadratic formula. Also, be sure to familiarize yourself with the discriminant and factoring trinomials, too.
The purpose of the rational zeroes theorem is not to find solutions to a polynomial, but to find potential solutions. That is, this root test really can only give us an idea at what some solutions could be. The actual solutions found by other methods could be all of them, some of them, or none at all! All this goes to show that the rational roots theorem is only one step in the greater journey to solve a polynomial. Once you’ve mastered other techniques, the rational root test becomes not very useful nor informative. One type of problem this theorem is useful, however, is when it comes to dealing with higher degree polynomials, which we will show in an example later.
Despite not being terribly useful for all types of problems, however, the rational root test is still a very important concept to understand! So, without further adieu, let’s learn how to try to find potential zeros of a function using the rational zero theorem.
How to Find the Zeroes of a Function:
The rational zero theorem is defined by the following formula:
That is, the rational root test utilizes the factors of the constant term (the term without a variable attached to it) and of the leading coefficient (usually with the highest degree variable) to find potential zeros. So, when we break it down like this, using the rational zero theorem is actually quite a simple process!
And of course, as always, the best way to learn this concept is to do an example problem.
Example:
Consider the following polynomial: $P(x) = 9x^{3} + 6x^{2}  29x  10$
a) Determine the Potential Zeros Using the Rational Root Theorem
Step 1: Identify the Coefficients and their Factors
If we recall the rational root theorem definition, we need to find the factors of both the constant term and the leading coefficient. In this case, the constant term is 10 and the leading coefficient is 9.
Factors of the constant term 10 = 1, 2, 5, 10
Factors of the leading coefficient 9 = 1, 3, 9
Step 2: Use Formula
Once we have identified the factors, all we need to do now is plug each of these factors in individually to get all of our potential solutions to the polynomial. This leaves use the the potential solutions of:
$\pm1, 2, 5, 10$
$\pm \frac{1}{3}, \frac{2}{3}, \frac{5}{3}, \frac{10}{3}$
$\pm \frac{1}{9}, \frac{2}{9}, \frac{5}{9}, \frac{10}{9}$
As you can see, there are many possible zeros. Now we must use other techniques to figure out which one(s), if any, are true solutions.
b) Factor P(x) fully
Since factoring cubic polynomials is difficult with the most commonly used methods, let’s factor using synthetic division. In order to use synthetic division, we will need to try all of the potential zeros found in part (a) of the question. Since there are so many, we will skip ahead a bit to x=2, which turns out to be the only zero above that is an actual zero of the polynomial.
Now, using synthetic division with x = 2:
So, $(x+2)$ is one of the factors of the polynomial. $(9x^{2}  12x  5)$ is also a factor. We can factor $(9x^{2}  12x  5)$ further using methods that we’re more familiar with, like the quadratic formula. After factoring, we have $(3x5)(3x+1)$ as the factors.
This gives us our final answer:
c) Solve the equation:
Since $(x + 2)(3x  5)(3x + 1)$ are the factors of $9x^{3} + 6x^{2}  29x  10$, we can solve $(x + 2)(3x  5)(3x + 1) = 0$ to easily get the answers to this part of the problem.
Solving each zero individually:
This gives us our final answers of:
And that’s all there is to it! For a quick check to see how well you’ve master this topic, check out this rational zero test calculator here to check your work. Lastly, for related exercise dealing with integration techniques, be sure to check out our video on integration by partial fractions.
Rational zeroes theorem
Basic concepts:
 Solving quadratic equations by completing the square
 Solving quadratic equations using the quadratic formula
 Nature of roots of quadratic equations: The discriminant
 Factoring trinomials: $ax^2 + bx + c$
Related concepts:
 Integration of rational functions by partial fractions
Lessons
Notes:
Rational Zero Theorem:

1.
Consider the polynomial $P\left( x \right) = 9{x^3} + 6{x^2}  29x  10$