Can a Non-Factorable Function Have Rational Real Roots?

[Miracles]

can a non-factorable function has rational real roots?

 

[Hootenanny]

can a non-factorable function has rational real roots?

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

 

[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).