$\displaystyle ax^3+bx^2+cx+d$ is a polynomial with integer coefficients. It is given that $ad,\,bc$ are odd and even respectively. Then prove that not all roots of the polynomial are rational.
It is easy to see that none of the roots are integer. But how to tackle the rational case? Any help is appreciated.
Hint $ $ If not, then it has $\,3\,$ rational roots of form $\,r/s\,$ for $\rm\color{#c00}{odd}$ $\,r,s,\,$ by the Rational Root Test. These rational roots persist modulo $2$ as roots $\,r/s\equiv \color{#c00}1/\color{#c00}1\equiv 1,\,$ therefore mod $\,2\,$ the polynomial $\equiv (x-1)^3\equiv x^3+x^2+x+1,\,$ thus $\,b\equiv 1\equiv c$ are both odd, contra $\,bc\,$ even.