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## Homework Statement

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Polynomial f(x) is divisible by ##x^2-1##. If f(x) is divided by ##x^3-x##, then the remainder is...

A. ##(x^2-x)f(-1)##

B. ##(x-x^2)f(-1)##

C. ##(x^2-1)f(0)##

D. ##(1-x^2)f(0)##

E. ##(x^2+x)f(1)##

## Homework Equations

Remainder theorem

## The Attempt at a Solution

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f(x) is divisible by ##x^2-1## which means

##f(x) = (x^2-1) H(x)+0 \\

f(x) = (x+1)(x-1) H(x) + 0 \\

f(1) = 0 \\

f(-1) = 0##

f(x) is divided by ##x^3-x## which means

##f(x) = (x^3-x) H(x) + (px+q) \\

f(x) = x (x^2-1) H(x) + px + q \\

f(x) = x(x+1)(x-1) H(x)+ px + q \\

\\

f(1) = p + q = 0 \\

f(-1) = -p+q = 0 \\

f(0) = q##

And, I got p = 0, and q = 0 which means no remainder for the division.

But, the options is very confusing.

Please help