Important: The polytools package has been deprecated. Use the superseding command PolynomialTools[IsSelfReciprocal] instead.

Determine whether a is a ``self-reciprocal'' polynomial in x. This property holds if and only if $\mathrm{coeff}\left(a\,x\,k\right)=\mathrm{coeff}\left(a\,x\,d-k\right)$ for all $k=0..d$, where $d=\mathrm{degree}\left(a\,x\right)$.

If d is even and if the optional second argument p is specified, p is assigned the polynomial P of degree $\frac{d}{2}$ such that $x^{\frac{d}{2}}P\left(x+\frac{1}{x}\right)=a$.

Note that if d is odd, a being self-reciprocal implies a is divisible by $x+1$. In this case, if p is specified then the result computed is for $\frac{a}{x+1}$.

$\mathrm{with}\left(\mathrm{polytools}\right)$

$\left[\mathrm{minpoly}\,\mathrm{recipoly}\,\mathrm{shorten}\,\mathrm{sort\_poly}\,\mathrm{split}\,\mathrm{splits}\,\mathrm{translate}\right]$

$\mathrm{recipoly}\left(x^4+x^3+x+1\,x\,'p'\right)$

$\mathrm{true}$

$p$

$x^2+x-2$

$\mathrm{recipoly}\left(x^5-3x^4+x^3+x^2-3x+1\,x\,'p'\right)$

$\mathrm{true}$

$p$

$x^2-4x+3$