The Rational Root Theorem

The rational root theorem states :

Let   P(x) = a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{1}x + a_{0}   be a polynomial with integer coefficients. Suppose that a = \frac{p}{q} is a root of   P(x)   where p and q have no common factors. Then a_{0} is divisible by p and a_{n} is divisible by q.

Since each of a_{n} and a_{0} have finitely many factors, for a given polynomial this theorem generates a finite list of rational numbers with the property that all rational roots of the polynomial are on the list. Thus, the rational root theorem gives a finite algorithm that can be used to find all rational roots of a given polynomial.

Some high school textbooks suggest that in order to find the roots of a polynomial with integer coefficients, one should use the rational root theorem to find all candidate rational roots of a polynomial equation and then plug them into the polynomial to determine whether they’re roots. The assumption implicit in this is that it will often happen that a polynomial with integer coefficients has a rational root. While this is often case for the high school textbook problems which have been contrived, this is seldom the case for general polynomials with integer coefficients and as such the textbook presentation of the application of the rational root theorem to find roots of a polynomial is misleading.

Polynomials with integer coefficients almost never have rational roots

Mathworld has a nice article with a picture illustrating which degree 5 polynomials x^5 + px + q with p, q integers factor further into polynomials with rational coefficients and which ones don’t; the former type corresponding to yellow boxes. From the picture it’s clear that the vast majority of polynomials of the type under consideration do not factor further into polynomials with rational coefficients. But if a polynomial does not factor further into polynomials with rational coefficients it can have no linear factors with rational coefficients so by the factor theorem it has no rational root.

The limited data illustrated is by no means a proof, however, the picture should be enough to dispel the notion that the rational root theorem provides a good method of finding the roots of a polynomial with integer coefficients. In fact, it has been proven that in a statistical sense, almost no polynomial with integer coefficients factors into polynomials with rational coefficients at all, this is even stronger than saying that a polynomial doesn’t have a linear factor with rational coefficients..

For practical applications one doesn’t need to find the exact values of the roots of a polynomial

It suffices to find a decimal approximation to the value of the root of a polynomial. While the number of decimals needed varies, according to Wikipedia

While the decimal representation of π has been computed to more than a trillion (1012) digits,[21] elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.

There are methods such as Newton’s method that almost always work to find very good decimal approximations to the roots of a polynomial. Newton’s method uses calculus but no more than the basics.

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2 thoughts on “The Rational Root Theorem

  1. This was a good article Dr. Sinick. When you mentioned these concepts in class I wanted to know more about how it all works. I found Newton’s Method very interesting. Keep posting articles.

  2. Pingback: The analogy between integers and polynomials | Advanced High School Mathematics

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