Rational root theorem
In algebra, the rational root theorem states that given an integer polynomial with leading coefficent and constant term , if has a rational root in lowest terms , and .
This theorem aids significantly at finding the "nice" roots of a given polynomial, since the coefficients entail only a finite amount of rational numbers to check as roots.
Proof
Let be a rational root of , where all are integers; we wish to show that and . Since is a root of , Multiplying by yields Using modular arithmetic modulo , we have . As and are relatively prime, . Via similar logic in modulo , , which completes the proof.
Problems
Here are some problems that are cracked by the rational root theorem. The answers can be found here.
Introductory
- Factor the polynomial .
Intermediate
- Find all rational roots of the polynomial .
- 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.