Find Polynomials Fulfilling Real Coefficient Equation

[anemone]

Let $k\ne 0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that $(x^3-kx^2+1)P(x+1)+(x^3+kx^2+1)P(x-1)=2(x^3-kx+1)P(x)$ for all real number $x$.

 

[anemone]

Spoiler: Official solution from the source of problem

Let $P(x)=a_nx^n+\cdots+a_0x^0$ with $a_n\ne 0$. Comparing the coefficients of $x^{n+1}$ on both sides gives $a_n(n-2m)(n-1)=0$ so $n=1$ or $n=2m$.

If $n=1$, one easily verifies that $P(x)=x$ is a solution, while $P(x)=1$ is not. Since the given condition is linear in $P$, this means that the linear solutions are precisely $P(x)=tx$ for $t\in \mathbb{R} $.

Now assume that $n=2m$. The polynomial $xP(x+1)-(x+1)P(x)=(n-1)a_nx^n+\cdots$ has degree $n$, and therefore it has at least one (possibly complex) root $r$. If $r\notin {0,\,-1)}$, define $k=\dfrac{P(r)}{r}=\dfrac{P(r+1)}{r+1}$. If $r=0$, let $k=P(1)$. If $r=-1$, let $k=-P(-1)$. We now consider the polynomial $S(x)=P(x)-kx$. It also satisfies the given equation because $P(x)$ and $kx$ satisfy it. Additionally, it has the useful property that $r$ and $r+1$ are roots.

Let $A(x)=x^3-mx^2+1$ and $B(x)=x^3+mx^2+1$. Plugging in $x=s$ into the given equation implies that

a. If $s-1$ and $s$ are roots of $S$ and $s$ is not a root of $A$, then $s+1$ is a root of $S$.

b. If $s$ and $s+1$ are roots of $S$ and $s$ is not a root of $B$, then $s-1$ is a root of $S$.

Let $a\ge 0$ and $b\ge 0$ be such that $r-a,\,r-a+1,\cdots,r,\,r+1,\cdots,\,r+b-1,\,r+b$ are roots of $S$, while $r-a-1$ and $r+b+1$ are not. The two statements above imply that $r-a$ is a root of $B$ and $r+b$ is a root of $A$.

Since $r-a$ is a root of $B(x)$ and of $A(x+a+b)$, it is also a root of their greatest common divisor $C(x)$ as integer polynomials. If $C(x)$ was a non-trivial divisor of $B(x)$, then $B$ would have a rational root $\alpha$. Since the first and last coefficients of $B$ are 1, $\alpha$ can only be 1 or -1, but $B(-1)=m>0$ and $B(1)=m+2>0$ since $n=2m$.

Therefore, $B(x)=A(x+a+b)$. Writing $c=a+b\ge 1$, we compute

$0=A(x+c)-B(x)=(3c-2m)x^2+c(3c-2m)x+c^2(c-m)$

Then we must have $3c-2m=c-m=0$, which gives $m=0$, a contradiction. We conclude that $f(x)=tx$ is the only solution.