Reciprocal Equations

Definition 3.1. A degree-$n$ polynomial $P(x)$ is a reciprocal polynomial

• of Type I if $P(x)=x^nP(1/x)$ — the equation $P(x)=0$ is then called a Type I reciprocal equation;
• of Type II if $P(x)=-x^nP(1/x)$ — a Type II reciprocal equation.

Theorem 3.6. For $a_nx^n+\cdots+a_1x+a_0=0$ ($a_n\ne0$), this is a reciprocal equation iff either $a_0=a_n,\ a_1=a_{n-1},\ a_2=a_{n-2},\ldots$ (Type I — coefficients from the start equal the coefficients from the end, in order) or $a_0=-a_n,\ a_1=-a_{n-1},\ a_2=-a_{n-2},\ldots$ (Type II — same coefficients from each end, but opposite in sign).

Proof idea. Replacing $x$ by $1/x$ in $P(x)=0$ and multiplying through by $x^n$ recovers the coefficients in reverse order. $P(x)=0$ is reciprocal iff this reversed polynomial is $\pm P(x)$ itself, which forces $a_r/a_{n-r}$ to be the same constant $\lambda$ for every $r$; multiplying the $r=0$ and $r=n$ instances of this ratio gives $\lambda^2=1$, so $\lambda=1$ (Type I, coefficients palindromic) or $\lambda=-1$ (Type II, coefficients anti-palindromic).