Rational Root Theorem

This post is a follow-up to the last post about the Fundamental Theorem of Algebra. As a recap, the Fundamental Theorem of Algebra says that if you have a polynomial of degree $n$ with coefficients in $\mathbb{C}$ , then the polynomial will have $n$ roots in $\mathbb{C}$ .

As we talked about last time, the Fundamental Theorem of Algebra is an existence proof because for a polynomial of degree $n$ whose coefficients are in $\mathbb{C}$ , it says that you’ll be able to find $n$ values which, when plugged into the polynomial, make the polynomial equal zero. In other words, you’re guaranteeing the existence of the values that will make the polynomial equal zero.

Existence proofs are cool and all, but the Fundamental Theorem of Algebra doesn’t tell us anything about how to find the $x-$ values that make the polynomial equal zero. Wouldn’t it be nice if there were a way to know where these zeros were?

It turns out that if we restrict ourselves to looking for rational zeros of a polynomial, then we can say exactly what those kinds of zeros would have to look like. Now, let’s state the Rational Root Theorem.

Rational Root Theorem: If a polynomial $f(x) = d_0x^n+d_1x^{n-1}+\ldots+d_{n-1}x+d_n$ has any rational zeros, then they will have to occur at $x=\frac{p}{q}$ , where $\pm p$ is some factor of $d_n$ and $\pm q$ is some factor of $d_0$ .

To prove the Rational Root Theorem, we’re going to write a polynomial as a product of terms, which the Fundamental Theorem of Algebra guarantees we can do.

Proof: Let $f(x) = (a_1+b_1x)(a_2+b_2x)\ldots(a_n+b_nx)$ , where the “ $\ldots$ ” includes the product of other linear terms. First, let’s find the zeros of this function. To do that, we want to solve $(a_1+b_1x)(a_2+b_2x)\ldots(a_n+b_nx) = 0$ . Note that we won’t always be able to get $n$ factors multiplied together if there are some factors that don’t yield rational roots. But even if you can’t, that gives you information, too! For example, there might be a polynomial with five roots, only three of which are rational.

Before we solve for the values of $x$ that make this function equal zero, let’s just think about what the highest degree term and lowest degree term will look like. We don’t need to think about the terms that have degree $0<k<n$ . The highest degree term comes from multiplying all of the terms that have an $x$ in them. And the lowest degree term comes from multiplying all of the terms that are just constants. This means that the highest degree term will be $(b_1)(b_2)\ldots (b_n)x^n$ . And the lowest degree term will be $(a_1)(a_2)\ldots (a_n)$ .

Let’s remember that we’re trying to solve $(a_1+b_1x)(a_2+b_2x)\ldots(a_n+b_nx) = 0$ . Since the left-hand side is a product of $n$ terms that equals zero, the whole left-hand side will equal zero when any of these terms individually equals zero. So, this function has zeros when $x\in\{-\frac{a_1}{b_1},-\frac{a_2}{b_2},\ldots,-\frac{a_n}{b_n}\}$ .

Okay, we’re so close! We know that if our function has a rational zero, it will definitely be in the set $A=\{-\frac{a_1}{b_1},-\frac{a_2}{b_2},\ldots,-\frac{a_n}{b_n}\}$ , even though not all of these quotients have to be rational. And if we want to find these values, we just have to search the set of quotients $\{\frac{p_i}{q_i}\}$ , where $p_i$ is a factor of $a_1a_2\ldots a_n$ and $q_i$ is a factor of $b_1b_2\ldots b_n$ because that will check all of the values in $A$ , which is the set of all zeros.

This theorem says that IF a function has rational zeros, then here’s how to find them. But there’s no guarantee that a function will have rational zeros to begin with.

So, to recap: The Fundamental Theorem of Algebra states the existence of $n$ complex-valued zeros for a polynomial of degree $n$ , but is kind of like “good luck!” when it comes to actually finding these zeros. Then the Rational Root Theorem comes along and says, “hey, if you don’t mind only looking for rational zeros of this function, I can help you out!” It says that any rational root of your polynomial can be written as the quotient of two integers, where the numerator is a factor of the constant term and the denominator is a factor of the highest degree term.

I hope this helped you see a connection between the Fundamental Theorem of Algebra and the Rational Root Theorem!

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Rational Root Theorem

Published by benjaminyellin6391

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