The IsSelfReciprocal(a, x) function determines 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 third 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 that a is divisible by $x+1$. In this case, if p is specified then the result computed is for $\frac{a}{x+1}$.

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

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

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

$\mathrm{true}$

$p$

$x^2+x-2$

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

$\mathrm{true}$

$p$

$x^2-4x+3$