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 $f(x)$.

If $(x-1)^3$ divides $f(x)+1$, then $(x-1)^2$ divides $f'(x)$.
If $(x+1)^3$ divides $f(x)-1$, then $(x+1)^2$ divides $f'(x)$.
As deg$\,f=5$, then deg$\,f'=4$, and hence $f'(x)=a(x-1)^2(x+1)^2=a(x^4-2x^2+1)$,
for some $a\in\mathbb R$.
Thus 
$$
f(x)=\frac{a}{5}x^5-\frac{2a}{3}x^3+ax+b,
$$
for some $b\in\mathbb R$.
Now, as $(x-1)^3$ divides $g(x)=f(x)+1$, in particular $f(1)+1=g(1)=0$. Thus
$$
-1=f(1)=\frac{a}{5}-\frac{2a}{3}+a+b. \tag{1}
$$ 
Similarly,
as $(x+1)^3$ divides $h(x)=f(x)-1$, in particular $f(-1)-1=g(-1)=0$. Thus
$$
1=f(-1)=-\frac{a}{5}+\frac{2a}{3}-a+b. \tag{2}
$$
Adding $(1)$ and $(2)$ we obtain that $b=0$, and thus $a=-15/8$.
Hence
$$
f(x)=-\frac{3}{8}x^5+\frac{5}{4}x^3-\frac{15}{8}x.
$$ 
