Let $f_1(x),f_2(x) \in K[x]$. Show that if $\alpha$ Is root of multiplicity $m \ge 1$ of the polynomial $$f_1(x)f_2’(x) - f_2(x)f_1’(x)$$ then $\alpha$ is a root of multiplicity $m+1$ of the polynomial $$f_1(\alpha)f_2(x) - f_2(\alpha)f_1(x)$$
In my attempt I could only see one thing from the first polynomial: $$= \Bigl(\frac{f_2(x)}{f_1(x)}\Bigr)’\cdot (f_1(x))^2$$ so $(x- \alpha)^m$ divides one of those two polynomials.
I’m lost here, I could use some help. Thanks.
