Properties of Polynomial Roots - Radical Conjugate Roots

Radical Conjugate Roots

It can be proved that if a polynomial P(x) with rational coefficients has a + √b as a root, where a, b are rational and √b is irrational, then a − √b is also a root. First observe that

Denote this quadratic polynomial by D(x). Then, by the Euclidean division of polynomials,

where c, d are rational numbers (by virtue of the fact that the coefficients of P(x) and D(x) are all rational). But a + √b is a root of P(x):

It follows that c, d must be zero, since otherwise the final equality could be arranged to suggest the irrationality of rational values (and vice versa). Hence P(x) = D(x)Q(x), for some quotient polynomial Q(x), and D(x) is a factor of P(x).

This property may be generalized as: If an irreducible polynomial P has a root in common with a polynomial Q, then P divides Q evenly.