<s>[INST] Solving polynomial equation with calculus I saw a question stating that 
Let $f(x)$ is a polynomial of degree $5 $.
$ x-1|f(x)+1 $ and $x+1|f(x)-1$ . Find $f(x)$ without using calculus.
And because I am not good at calculus , I was not able to solve the question with or without using calculus.
Please help me in solving it.
A hint for how to start could work.
 [/INST] Followup on previous comment:

The two conditions are equivalent to $f(1)=−1$ and $f(−1)=1$ but that's not enough to determine the polynomial univocally.

Let the quintic be $f(x) = ax^5+bx^4+cx^3+dx^2+ex+f$, then the two conditions give:
$$
\begin{align}
\begin{cases}
a+b+c+d+e+f & = -1 \\
-a+b-c+d-e+f &= 1
\end{cases}
\end{align}
$$
Adding and subtracting the two equations:
$$
\begin{align}
\begin{cases}
a+c+e &= -1 \\
b+d+f & = 0
\end{cases}
\end{align}
$$
The above give the necessary and sufficient relations that the coefficients of $f(x)$ must satisfy, but do not determine $f(x)$ univocally. For example, all of the following satisfy the given conditions:


*

*$f(x) = -x^5$ 

*$f(x) = x^5 - x^3 - x$

*$f(x) = x^5 + 3 x^4 - x^3 - 5 x^2 - x + 2$

*$f(x) = a x^5 + b x^4 + cx^3 +d x^2 - (a+c+1)x - b-d \quad \text{for} \;\forall a,b,c,d$
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