Let $f(x)$ and $g(x)$ be polynomials in $\mathbb{Q}[x]$ such that $f(x)$ is irreducible in $\mathbb{Q}[x]$ and $g(x)\neq0$. If $\alpha \in \mathbb{R}$ exists such that $f(\alpha)=g(\alpha)=0$, meaning that, $f(x)$ and $g(x)$ have a common root in $\mathbb{R}$, show that $f(x) \mid g(x)$ in $\mathbb{Q}[x]$
I dont´n see how $f(x) \mid g(x)$ is true only by knowing that they have a common root in $\mathbb{R}$. Could anyone give me a hint?
Hint Let $h(x)$ be the greatest common divisor of $f(x)$ and $g(x)$ in $\mathbb{Q}[x]$.
By the extended Euclidian Algorithm you have $$h(x)=P(x)f(x)+Q(x)g(x) \Rightarrow h(\alpha) =0 0$$
This implies that $\deg(h) \geq 1$.
Since $\deg(h) \geq 1, h|f$ and $f$ is irreducible, we must have $$f(x)=ch(x)$$ for some constant $c$. Since $h|g$ deduce that $f$ divides $g$.