# Rational Root Theorem

Given a polynomial with integral coefficients, . The **Rational Root Theorem** states that if has a rational root with relatively prime positive integers, is a divisor of and is a divisor of .

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.

## Contents

## Proof

Given is a rational root of a polynomial , where the 's are integers, we wish to show that and . Since is a root, Multiplying by , we have: Examining this in modulo , we have . As and are relatively prime, . With the same logic, but with modulo , we have , which completes the proof.

## Problems

### Easy

1. Factor the polynomial .

### Intermediate

2. Find all rational roots of the polynomial .

3. Prove that is irrational, using the Rational Root Theorem.

### Answers

1.

2.

3. A polynomial with integer coefficients and has a root as must also have as a root. The simplest polynomial is which is . We see that the only possible rational roots are and , and when substituted, none of these roots work.
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