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I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that:

$f_{1}(p_{1}) = f_{2}(p_{1})$.

Let $\alpha\in(0,1]$ be a constant. I want to know general rules on the coefficients of $f_{1}(p)$ and $f_{2}(p)$ such that the polynomial equation:

$\alpha\cdot f_{1}(p) = f_{2}(p)$

also has a root in $(0,1)$.

Moreover, I could also assume that: For any $\alpha$, if at all there is a root to the above polynomial equation, then there is only one root. (I mean that $\alpha\cdot f_{1}(p)$ and $f_{2}(p)$ cross each other either only once or they don't cross at all, in the range $p\in(0,1)$.

Could anyone please tell me what theorems might be relevant for me.

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