Let $f:\mathbb{R} \to \mathbb{R}$ be a polynomial function with rational coefficients and degree $3$. If the graphic of $f$ is tangent to the $x$ axis. Show that all zeros of $f$ are rational numbers.
If $\alpha$ is a zero of $f$, i.e., if $f(\alpha)=0$ and the graph of $f$ is tangent to the $x$ axis in this point, then $\alpha$ has even multiplicity, but I don't know how I can conclude this question. Someone can help-me?
Thanks a lot.
Best Answer
If $\alpha$ is a root where the graph of $f$ is tangent to the $x$ axis, then $\alpha$ is a root of multiplicity $\ge 2$ (not necessarily even).
Case 1: $\alpha$ has multiplicity $3$. Then $\alpha$ is a root of the polynomial $f''(x)$ which has degree $1$ and rational coefficients, and therefore $\alpha$ is rational.
Case 2: $\alpha$ has multiplicity $2$. Then the gcd of $f(x)$ and $f'(x)$ is $x-\alpha$. But the gcd of two polynomials with rational coefficients has rational coefficients, so $\alpha$ must be rational.