# Difference between revisions of "Rational Root Theorem"

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==Problems== | ==Problems== | ||

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+ | Factor the polynomial <math>x^3-5x^2+2x+8</math>. | ||

===Intermediate=== | ===Intermediate=== |

## Revision as of 21:04, 1 February 2012

*This article is a stub. Help us out by expanding it.*

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 , 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 , and we are done.

## Problems

### Easy

Factor the polynomial .

### Intermediate

Find all rational roots of the polynomial .

Prove that is irrational, using the Rational Root Theorem.