8.3 RATIONAL ROOTS

We now consider a polynomial equation
 

a₀xⁿ + a₁x^{n -1} + ... + a_{n -1}x + a_n = 0, 

of degree n with integer coefficients a_i. It can be shown that if there is a rational root p/q, with p and q integers having no common integral divisor greater that 1, then p must be an integral divisor of a_n and q must be an integral divisor of a₀. For example, if the rational number p/q in lowest terms is a root of
 

then p must be one of the twelve integral divisors of -12 and q one of the integral divisors of 6. Without losing any of the possibilities, we may restrict q to be positive, that is, to be one of the integers 1, 2, 3, 6. The possible rational roots, therefore, are

Trials would show that 3/2 and -4/3 are the only rational roots.

Example. Prove that is not a rational number.

Solution: Let  Then

Hence a is a root of x⁴ - 10x² + 1 = 0. This fourth degree polynomial equation has integer coefficients. The rule on rational roots tells us that the only possible rational roots are 1 and -1. Substituting, we see that neither 1 nor -1 is a root. Hence there are no rational roots. Since a is a root, it follows that a is not rational.