Mathematical Reflections 2014, Issue 4 - Problem U307

Problem:
Prove that any polynomial $f \in \mathbb{R}[X]$ can be written as a difference of increasing polynomials.

Proposed by Jishnu Bose.

Solution:
Let $f(X)=a_nX^n+a_{n-1}X^{n-1}+\ldots+a_1X+a_0$. We construct the following algorithm. Set $g_0(X)=a_0, h_0(X)=0$ and for $k=1,\ldots,n$, if $k$ is odd, then $$g_k(X)=\begin{cases} a_kX^k & \textrm{if } a_k \geq 0 \\ 0 & \textrm{if } a_k<0, \end{cases}$$ $$h_k(X)=\begin{cases} 0 & \textrm{if } a_k \geq 0 \\ -a_kX^k & \textrm{if } a_k<0, \end{cases}$$ and if $k$ is even, then $$g_k(X)=\begin{cases}a_k(X^{k+1}+X^k+X^{k-1}) & \textrm{if } a_k \geq 0 \\ -a_k(X^{k+1}+X^{k-1}) & \textrm{if } a_k < 0,  \end{cases}$$ $$h_k(X)=\begin{cases}a_k(X^{k+1}+X^{k-1}) & \textrm{if } a_k \geq 0 \\ -a_k(X^{k+1}+X^k+X^{k-1}) & \textrm{if } a_k < 0. \end{cases}$$
Observe that for all $k=0,\ldots,n$, we have $g_k(X)-h_k(X)=a_kX^k$ and $g'_k(X) \geq 0$, $h'_k(X) \geq 0$ for all $x \in \mathbb{R}$. Setting $$g(X)=\sum_{k=0}^n g_k(X), \qquad h(X)=\sum_{k=0}^n h_k(X),$$ we have that $g(X)$ and $h(X)$ are increasing polynomials and $f(X)=g(X)-h(X)$.