TITLE: Finding polynomial.
QUESTION [5 upvotes]: Let $f(x)$ be a fifth degree polynomial. Such that$(x-1)^3$ divides $(f(x)+2)$
&$(x+1)^3$ divides $(f(x)-2)$.
Find $f(x)$.
It was to be find without using calculus.but l don't know how to do this .
I was surprised that how it can be done using calculus.
Please tell me how to solve such questions
I would be thankful.

REPLY [3 votes]: Hint: As $(x-1)^3$ divide $f(x)+2$, we have that $(x-1)^2$ divide $f^{\prime}(x)$. Also, $(x+1)^2$ divide $f^{\prime}(x)$. As $f$ is of degree $5$, there exists a constant $c$ such that $f^{\prime}(x)=c(x-1)^2(x+1)^2=c(x^2-1)^2$. I think you can take it from here...