# Polynomial Long division

## Homework Statement

$$\frac{x^5-a^5}{x^2-a^2}$$, where a is some constant.

## The Attempt at a Solution

I can't figure out how to do this with long division. With synthetic, I can get to $$\frac{a^4+a^3 x+a^2 x^2+a x^3+x^4}{a+x}$$

Code:
                  x^3+xa^2+?
_______________
x^2-a^2      ) x^5-a^5
-x^5 + x^3a^2
---------------
0-  a^5+x^3a^2
xa^4-x^3a^2
---------------
Hopefully it's possible to decipher my steps from that diagram, I don't know how to write long division in latex. I'm left with -a^5+xa^4, which doesn't go evenly into the divisor. I thought about writing $$\frac{-a^5+xa^4}{x^2-a^2}$$ in place of the ? mark, but the entire purpose of this is to take the limit as x->a, and I would be left with division by zero. Surely this isn't only solvable by synthetic division?

Edit: For prosperity's sake, the limit is (5a^3)/2

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vela
Staff Emeritus
Homework Helper
What you did so far is fine. You can simplify your expression for the remainder. Factor ##a^4## out of the numerator, and factor the denominator. You'll get some cancellation that will allows you to evaluate the limit.

I'm not seeing the cancellation. Could you elaborate?

vela
Staff Emeritus