A monic polynomial P(x) of degree 3 satisfies the following conditions:
$\displaystyle x-1$ divides P(x)+1 exactly
$\displaystyle x+1$ divides P(x)-1 exactly
and P(0)=1
Determine P(x)
Your polynomial will be in the form: $\displaystyle x^3 + ax^2 + bx + c$
$\displaystyle P(x) = x^3 + ax^2 + bx + c$
$\displaystyle P(0) = (0)^3 + a(0)^2 + b(0) + c = 1 \rightarrow c=1$
$\displaystyle P(x)+1=0$ when factor $\displaystyle (x-1)$ is divided hence root is $\displaystyle 1$. Therefore, $\displaystyle P(1) + 1 = (1)^3 + a(1)^2 + b(1) + 1 + 1 = 0 \rightarrow a+b+2 = 0$.
$\displaystyle P(x)-1=0$ when factor $\displaystyle (x+1)$ is divided hence root is $\displaystyle -1$. Therefore, $\displaystyle P(-1) + 1 = (-1)^3 + a(-1)^2 + b(-1) + 1 + 1 = 0 \rightarrow a-b+2 = 0$.
You have 2 simultaneous equation. Just got to solve for $\displaystyle a,b$ and you will have your monic polynomial $\displaystyle P(x)$.