Rational roots

1. Feb 14, 2007

Miracles

can a non-factorable function has rational real roots?

Last edited: Feb 14, 2007
2. Feb 14, 2007

Hootenanny

Staff Emeritus
Yes, for example take $f(x):= x+a$ where $a \in \mathbb{Q}$.

3. Feb 14, 2007

StatusX

Are you asking if a polynomial with rational coefficients which cannot be written as the product of polynomials with rational coefficients of smaller degree can have rational roots? Except for the trivial case of polynomials of degree one, the answer is no, because any rational root a of f(x) translates to a linear factor (x-a) of f(x), ie, there is a polynomial g(x) with rational coefficients such that f(x)=(x-a)g(x).