Reciprocal Equations

The fifth standard transformation replaces each root by its reciprocal: if $\alpha$ is a root of the degree-$n$ equation $f(x)=0$, then $1/\alpha$ is a root of $x^nf(1/x)=0$, which amounts simply to writing the original coefficients in reverse order.

An equation is called a reciprocal equation if this transformation returns essentially the same equation -- equivalently, if $1/\alpha$ is a root whenever $\alpha$ is. This happens exactly when the coefficients, read from either end, match: class one, where $a_k=a_{n-k}$ for every $k$ (symmetric, same sign), or class two, where $a_k=-a_{n-k}$ for every $k$ (symmetric, alternating sign).

For an equation of even degree $n$, dividing through by $x^{n/2}$ and grouping terms symmetric about the centre expresses the whole left side as a polynomial in $t=x+\dfrac1x$ (class one) or in terms built from $x-\dfrac1x$ (class two), using identities such as $x^2+\dfrac1{x^2}=t^2-2$. This turns a degree-$n$ problem into one of degree $n/2$ in $t$; each value of $t$ found is then turned back into two values of $x$ by solving $x^2-tx+1=0$. …