What is the Remainder When f(x) is Divided by (x+1)?

[songoku]

Homework Statement

Given that $f(x)$ is divisible by $(x-1)$, then the remainder when $f(x)$ is divided by $(x-1)(x+1)$ is
a. $- \frac{f(-1)}{2} (1+x)$
b. $- \frac{f(-1)}{2} (1-x)$
c. $\frac{f(-1)}{2} (1+x)$
d. $\frac{f(-1)}{2} (1-x)$
e. $\frac{f(-1)}{2} (x-1)$

Relevant Equations

polynomial

$f(x)$ is divisible by $(x-1) \rightarrow f(1) = 0$ $f(x) = Q(x).(x-1)(x+1) + R(x)$ where $Q(x)$ is the quotient and $R(x)$ is the remainderSeeing all the options have $f(-1)$, I tried to find $f(-1)$:
$f(-1) = R(-1)$

I do not know how to continue

Thanks

 

[PeroK]

Why not try some examples for $f(x)$ and see what happens?

 

[haruspex]

What can R look like? Max degree, e.g.?

 

[Delta2]

This formula you wrote $$f(x)=Q(x)(x-1)(x+1)+R(x)$$ says a lot if you know how to interpret it. Since (x-1) divides f(x) what can you say about whether (x-1) divides R(x)?
I believe this and together with the answer to the question of @haruspex will get you to the answer of the question.

 

[FactChecker]

Why not try some examples for $f(x)$ and see what happens?

In particular, what happens for $f(x) = (x-1)$?

 

[PeroK]

In particular, what happens for $f(x) = (x-1)$?

Yes, that's not bad place to start.

 

[songoku]

Thank you very much for the help PeroK, haruspex, Delta2, Factchecker

 

[ehild]

It seems that the problem is solved.. so I can show a solution in detail.
f(x) is divisible by (x-1):
f(x)=g(x)(x-1)...(1)
g(x) divided by (x+1), the remainder is g(-1):.
g(x)=p(x)(x+1)+g(-1)...(2)
So f(x)=(p(x)(x+1)+g(-1))(x-1)=p(x)(x+1)(x-1)+g(-1)(x-1)...(3)
f(x) divided by (x+1) the remainder is f(-1) . From (3)
f(-1)=g(-1)(-2)...(4)
f(x) divided by (x-1)(x+1), the remainder is g(-1)((x-1) . Because of (4), this is equal to f(-1)/(-2)(x-1) =f(-1)(1-x)/2