Translate the polynomial $a\left(x\right)$ by $x=x+\mathrm{x0}$ efficiently. The method used requires $\mathrm{O}\left(n\right)$ multiplications and divisions and $\mathrm{O}\left(n^2\right)$ additions where $n=\mathrm{degree}\left(a\,x\right)$.

This function is part of the PolynomialTools package, and so it can be used in the form Translate(..) only after executing the command with(PolynomialTools). However, it can always be accessed through the long form of the command by using PolynomialTools[Translate](..).

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\:$

$\mathrm{Translate}\left(x^2\,x\,1\right)$

$x^2+2x+1$

$\mathrm{expand}\left(\mathrm{eval}\left(x^2\,x=x+1\right)\right)$

$x^2+2x+1$

$\mathrm{Translate}\left(x^3\,x\,2\right)$

$x^3+6x^2+12x+8$

$\mathrm{expand}\left(\mathrm{eval}\left(x^3\,x=x+2\right)\right)$

$x^3+6x^2+12x+8$

$\mathrm{Translate}\left({\left(x+1\right)}^3\,x\,-1\right)$

$x^3$