# Rational root theorem

In algebra, the **rational root theorem** states that given an integer polynomial with leading coefficient and constant term , if has a rational root in lowest terms, then and .

This theorem is most often used to guess the roots of polynomials. It sees widespread usage in introductory and intermediate mathematics competitions.

## Proof

Let be a rational root of , where every is an integer; we wish to show that and . Since is a root of , Multiplying by yields Using modular arithmetic modulo , we have , which implies that . Because we've defined and to be relatively prime, , which implies by Euclid's lemma. Via similar logic in modulo , , as required.

## Examples

Here are some problems with solutions that utilize the rational root theorem.

### Example 1

*Find all rational roots of the polynomial .*

**Solution**: The polynomial has leading coefficient and constant term , so the rational root theorem guarantees that the only possible rational roots are , , , , , , , and . After testing every number, we find that none of these are roots of the polynomial; thus, the polynomial has no rational roots.

### Example 2

*Factor the polynomial .*

**Solution**: After testing the divisors of 8, we find that it has roots , , and . Then because it has leading coefficient , the factor theorem tells us that it has the factorization .

### Example 3

*Using the rational root theorem, prove that is irrational.*

**Solution**: The polynomial has roots . The rational root theorem guarantees that the only possible rational roots of this polynomial are , and . Testing these, we find that none are roots of the polynomial, and so it has no rational roots. Then because is a root of the polynomial, it cannot be a rational number.