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 $f(x)$.

We can clearly see that: $f(1)+1=0$ and $f(-1)-1=0$
We can write $f(x)-1=p(x)(x+1)^3$ and $f(x)+1=q(x)(x-1)^3$
By differentiation and double differentiation, you can see that
$f'(1)=0$ and $f''(1)=0$
AND 
$f'(-1)=0$ and $f''(-1)=0$
You got six conditions and six unknowns! 
[assume $f(x) = x^6+a_1x^5+a_2x^4+a_3x^3+a_4x^2+a_5x+a_6$]
