Rational Root Theorem
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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 , 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.
Factor the polynomial .
Find all rational roots of the polynomial .
Prove that is irrational, using the Rational Root Theorem.