Basic Transformations: Sign, Scale and Translation

Four elementary substitutions cover most of what is needed:

Change of sign. If $\alpha$ is a root of $f(x)=0$, then $-\alpha$ is a root of $f(-x)=0$; simply substitute $-x$ for $x$ throughout (even-power terms are unaffected, odd-power terms change sign).

Multiplying the roots by $k$. If $\alpha$ is a root of $f(x)=0$, then $k\alpha$ is a root of $f(x/k)=0$; substitute $x/k$ for $x$ and clear the resulting denominators.

Translating the roots (diminishing or increasing by $h$). If $\alpha$ is a root of $f(x)=0$, then $\alpha-h$ is a root of the equation obtained by expanding $f(y+h)$ and setting it to zero in the new variable $y=x-h$. Rather than expanding directly, this is done efficiently by Horner's process: divide $f(x)$ synthetically by $(x-h)$, then divide the quotient by $(x-h)$ again, and again, until only a constant remains. The successive remainders produced -- read from the last one obtained back to the first -- are exactly the coefficients of the new equation in $y$, constant term first. A particularly useful choice is $h=-p_1/n$ for the monic degree-$n$ equation $x^n+p_1x^{n-1}+\cdots$: this removes the second-highest ($x^{n-1}$) term of the equation entirely, a standard first step before tackling a cubic or quartic by other methods. …