<s>[INST] Find polynomial $f(x)$ based on divisibility properties of $f(x)+1$ and $f(x) - 1$ $f(x)$ is a fifth degree polynomial. It is given that $f(x)+1$ is divisible by $(x-1)^3$ and $f(x)-1$ is divisible by $(x+1)^3$. Find $f(x)$.
 [/INST] Note that $(x-1)^3$ divides $f(x)+1$ and $f(-x)-1$, so $(x-1)^3$ divides the sum $f(x)+f(-x)$.
Similarly, we can note that $(x+1)^3$ divides $f(x)+f(-x)$.
Therefore, $(x+1)^3(x-1)^3$ divides $f(x)+f(-x)$, which has degree at most $5$.
This implies that $f(x)$ is an odd function.
We get $f(x)+1=(x-1)^3(ax^2+bx-1)$, (note that $f(0)$ must be $0$) and by comparing the degree $2$ and $4$ coefficients (which should be $0$), we get $a=-\frac{3}{8}$ and $b=-\frac{9}{8}$, which gives the answer of $$f(x)=-\frac{3}{8}x^5+\frac{5}{4}x^3-\frac{15}{8}x$$
</s>