Let $p(x)=x^4+ax^3+bx^2+cx+d$ be a polynomial such that there exists only one $r\in\mathbb{R}$ such that $p(r)=0.$ If $a,b,c,d\in\mathbb{Q},$ show that $r\in\mathbb{Q}.$
I do not know how to approach this question. In a hint, we were asked to examine the roots of $p'$ as well. Well, if $r$ is the only real root of a quartic polynomial, it must occur with multiplicity $2$ or $4.$ If it occurs with multiplicity $4,$ then, $p'''(r)=24r+6a=0.$ This gives $r=\frac{-a}{4},$ which is rational. How to deal with the multiplicity being $2$ case?
PS: This was a question on an entrance exam that ended $4$ hours ago.