Factoring polynomials has always been a challenge for many students, particularly high school students. The good news is that there are various methods to factoring polynomials ,and one of the most useful is the Rational Zeros Theorem.
The Rational Zeros Theorem is for solving polynomial equations, which helps us find the rational zeros of a polynomial equation. Keep reading to find out more about the Rational Zeros Theorem’s definition, its importance, and how to apply it to solve polynomial equations.
If you wonder how to do Long Division with Polynomials, I encourage you to check out this article where I share easy steps by step.
What is The Rational Zeros Theorem?
The Rational Zeros Theorem tells us that if a polynomial has integer coefficients, its rational roots will always be in the form of a fraction ±p/q where p is a factor of its constant term and q is a factor of its leading coefficient.
For example, a polynomial equation can be written in the form of axn + bx(n-1) + … + kx + l = 0, where x is the variable, and a, b, c, … k, l are the coefficients. And the Rational Zeros Theorem states that any rational roots of a polynomial equation can be written in the form of ±p/q, where p divides the constant term l (the coefficient of x0), and q divides the leading coefficient a (the coefficient of xn).
An an illustration, if a polynomial has the equation f(x) = f(x) = x3-6x2+11x-6 we can determine all of its rational roots (if it has any) by listing all of the factors of 1 and 6 as well as their corresponding combinations.
I encourage you to watch this video to learn how to use the rational zero test to determine all possible rational zeros.
Example Of How The Rational Zeros Theorem Works?
Let’s illustrate how the Rational Zeros Theorem works. If we have a polynomial equation f(x) = x3-6x2+11x-6, we can use the theorem to find all of its rational roots (if any):
- First, start by listing all of the factors of the constant term, which is -6: ± 1, ± 2, ± 3, and ± 6.
- Second, list all of the factors of the leading coefficient, which is 1: ± 1.
- And then find all of the possible rational roots: ± 1/1, ± 2/1, ± 3/1, and ± 6/1. These are also known as the potential rational roots (PRR).
After we we have all the possible rational roots, we can replace each possible rational root to find out which one is actual zero of the function.
For example,
- f(-1) = – 24, which means that – 1 is not a zero of f(x) = x3-6x2+11x-6
- But 1 is one of the zeros of f(x) = x3-6x2+11x-6 because f(1) = 0
Now we can use synthetic division or polynomial long division to find out which the rest of zeros of the function. In this example, x = 1, x = 2, and x = 3 are the three rational roots of the polynomial equation.
When Is The Rational Zeros Theorem Useful?
The Rational Zeros Theorem is most effective when dealing with polynomials that have high degrees. I believe that Rational Zeros theorem can save you a lot of time while factoring a polynomial equation and reduce your workload.
Apart from high degrees, when you have an irrational number as a root of an equation, you can use Rational Zeros Theorem to find its rational zeros.
I encourage you to watch the video below to explore how to find all the zeros of a polynomials function.
Advantages of Rational Zeros Theorem
Rational Zeros Theorem helps us to find out whether a polynomial has rational zeros ( also known as roots. The Rational Zeros Theorem is useful in helping us quickly find a complete list of possible rational roots of the polynomial.
Moreover, the Rational Zeros Theorem is beneficial when we need to factor a polynomial using polynomial long division or synthetic division.
In other words, the Rational Zeros Theorem gives an easy method for finding the roots of a polynomial expression with integer coefficients. Instead of randomly trying factors, Rational Zeros Theorem helps you find possible zeros of polynomials functions of give you a heads-up in proceeding with polynomial long division or synthetic division.
To apply the Rational Zeros Theorem, we must first identify the possible rational roots of the polynomial equation. We can use the theorem to list the possible rational roots of a polynomial equation and then use synthetic division or long division to determine which of these roots are actually the roots of the equation.
Typically, the theorem’s significance lies in the fact that it makes finding the rational roots of a polynomial equation more streamlined and efficient.
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- Tangent Line: What a tangent line is, how to find it, and why it matters?
- Solving Systems of Equations by the Elimination Method.
- IB vs. AP Classes: A Breakdown of Pros and Cons.
- 9 Best YouTube Channels to Learn Calculus.
Wrapping Up
The Rational Zeros Theorem helps you save time and simplifies factoring polynomials. It is an essential tool in algebra that helps students, particularly AP calculus and A Level mathematics students solve complex equations with ease.
I believee that by using the possible rational roots identified by the Rational Zeros Theorem, you can eliminate the guesswork and narrow down your search for the roots of the equation.