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Let $p(x)$ be a degree $d$ polynomial such that $|p(x)|\leq 1$ for $x \in [-1,1]$. Assume I have constructed a quantum circuit, $V(x)$, that implements $p(x)$ in the upper left entry of a $2$ by $2$ unitary matrix by quantum signal processing.

I now want to implement the polynomial $\alpha p(x)$ for $\alpha$ such that $|\alpha| \leq 1$. Is it possible to implement $\alpha p(x)$ using hopefully not too complicated manipulations to $V(x)$?

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