If $c$ is a root of multiplicity $r_{1}$ in a polynomial $p_{1}(x)$ and $c$ is a root of multiplicity $r_{2}$ in a polynomial $p_{2}(x)$. Show that $c$ is a root of multiplcity $\leq$ than $min(r_{1},r_{2})$ of the polynomial $p_{1}(x)q_{1}(x)+p_{2}(x)q_{2}(x)$ does not matter which polynomials are $q_{1}(x)$ and $q_{2}(x)$.
I see of course that $c$ is a root of the polynomial $p_{1}(x)q_{1}(x)+p_{2}(x)q_{2}(x)$ so then I was thinking using the factorization theorem every real polynomial can be seen as $f(x)=\lambda(x-\alpha_{1})...(x-\alpha_{2})$ where is $\lambda$ is a constant not zero. So this makes me see that in the factorization of$p_{1}$ $(x-c)$ appears $r_{1}$ times and it appears $r_{2}$ times in $p_{2}$, by applying the same factorization theorem in the polynomial $p_{1}(x)q_{1}(x)+p_{2}(x)q_{2}(x)$ the linear factor $(x-c)$ will apear $r_{1}+r_{2}$ times and this is of course a multiplcity bigger or equal than $r_{1}$ and $r_{2}$ by separateso it be bigger or equal than $min(r_{1},r_{2})$. Is my argument right?
Also, why if $r_{1} > r_{2}$ then $c$ is a root with multiplicity $r_{2}$ in in $p_{1}+p_{2}$?
Thanks :)