Transform a polynomial so that positive roots are shifted right and negative roots are shifted left I'm trying to figure out if it is possible to shift the roots of a polynomial outward, instead of to the left or right.  Its relatively simple to shift all the solutions in one direction by substituting (x-k) or (x+k) for x within the equation, but is it possible to shift all the positive solutions to the right and all the negative solutions to the left?  

Let $p(z) = a_0 + a_1z + a_2 z^2 + \dots + a_m z^m$ and let $z_1, \dots, z_m \in \mathbb{C}$ be the roots of $p$. For $\lambda \in \mathbb{C} \setminus \{0\}$, set $q(z) = b_0 + b_1z + \dots + b_mz^m$, where $b_j = \lambda^{-j} a_i$. Then the zeros of $q$ are $\lambda z_1, \dots, \lambda z_m$. 
Therefore if $\lambda > 0$, all roots are moved away from $0$ and if $0 < \lambda < 1$, all roots are moved closer to $0$. For $\lambda = -1$, negative roots become positive and positive roots become negative.   
