Q:

How to show that polynomial equation?

Let $q(z)$ be a polynomial function, where $z$ is any complex number. $q(z)$ could be written as $q(z)=a_0+a_kz^k+z^{k+1}h(z)$, where $a_k \neq 0$ is the first nonzero coefficient after $a_0$ and $h(z)$ is a polynomial of degree $n-k-1$.
Show that for $t>0$, $$q(tw)=(1-t^k)a_0+t^{k+1}w^{k+1}h(tw)$$ where $w$ is the $k$th root of $-a_0/a_k$, $w^k=-a_0/a_k$.

A:

Let $z=tw$, then we obtain
$$q(tw)=a_0+a_k(tw)^k+(tw)^{k+1}h(tw)\\
=a_0+a_kt^k(-a_0/a_k)+t^{k+1}w^{k+1}h(tw)\\
=a_0-a_ot^k+t^{k+1}w^{k+1}h(tw)\\
=(1-t^k)a_0+t[t^kw^{k+1}h(tw)].$$

