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).

Since $$f(x)+1$$ vanishes to third order at $$x=1$$ and $$f(x)-1$$ vanishes to third order at $$x=-1$$, differentiating both conditions shows $$f'(x)$$ must be a multiple of $$(x-1)^2(x+1)^2 = (x^2-1)^2$$. Writing $$f'(x) = \lambda(x^4-2x^2+1)$$ and integrating gives $$f(x) = \lambda\left(\frac{x^5}{5}-\frac{2x^3}{3}+x\right)+C$$, and applying $$f(1)=-1$$ and $$f(-1)=1$$ solves for $$C=0$$ and $$\lambda=-\frac{15}{8}$$. This gives $$f(x) = -\frac{3}{8}x^5+\frac{5}{4}x^3-\frac{15}{8}x$$.