Rational root theorem
This theorem is most often used to guess the roots of polynomials. It sees widespread usage in introductory and intermediate mathematics competitions.
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.
Intro to Rational Roots theorem: https://www.youtube.com/shorts/wKpmfnyKeeM
Here are some problems with solutions that utilize the rational root theorem.
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.
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 .
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.