Rational Root Theorem
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.
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.
1. Factor the polynomial .
2. Find all rational roots of the polynomial .
3. Prove that is irrational, using the Rational Root Theorem.
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. This article is a stub. Help us out by.