Rational Root Theorem

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Given a polynomial $P(x) = a_n x^n + a_{n - 1}x^{n - 1} + \ldots + a_1 x + a_0$ with integral coefficients, $a_n \neq 0$ . The Rational Root Theorem states that if $P(x)$ has a rational root $r = \pm\frac pq$ with $p, q$ relatively prime positive integers, $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$ .

As a consequence, every rational root of a monic polynomial with integral coefficients must be integral.

This gives us a relatively quick process to find all "nice" roots of a given polynomial, since given the coefficients we have only a finite number of rational numbers to check.

Proof

Given $\frac{p}{q}$ is a rational root of a polynomial $f(x)=a_nx^n+x_{n-1}x^{n-1}+\cdots +a_0$ , where the $a_n$ 's are integers, we wish to show that $p|a_0$ and $q|a_n$ . Since $\frac{p}{q}$ is a root, $0=a_n\left(\frac{p}{q}\right)^n+\cdots +a_0$ Multiplying by $q^n$ , we have: $0=a_np^n+a_{n-1}p^{n-1}q+\cdots+a_0q^n$ Examining this in modulo $p$ , we have $a_0q^n\equiv 0\pmod p$ . As $q$ and $p$ are relatively prime, $p|a_0$ . With the same logic, but with modulo $q$ , we have $q|a_n$ , which completes the proof.

Problems

Easy

1. Factor the polynomial $x^3-5x^2+2x+8$ .

Intermediate

2. Find all rational roots of the polynomial $x^4-x^3-x^2+x+57$ .

3. Prove that $\sqrt{2}$ is irrational, using the Rational Root Theorem.

Answers

1. $(x-4)(x-2)(x+1)$

2. $\text{There are no rational roots for the polynomial.}$

3. A polynomial with a root as $\sqrt{2}$ must also have $-\sqrt{2}$ as a root. The simplest polynomial is $(x+\sqrt{2})(x-\sqrt{2})$ which is $x^2-2=0$ . We see that the only possible rational roots are $\pm 1$ and $\pm 2$ , and when substituted, none of these roots work.

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