Difference of Two Odd Powers/Mistake

Source Work

 * Chapter $2$: Special Products and Factors

This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).

Mistake

 * $x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \paren {x^{n - 1} + x^{n - 2} y + x^{n - 3} y^2 + \dotsb} \paren {x^{n - 1} - x^{n - 2} y + x^{n - 3} y^2 - \dotsb}$

Correction
This result is true only if $n$ is an odd integer.

When $n$ is even, we get:

and there is no obvious way to factorise $\displaystyle \sum_{j \mathop = 0}^{n - 1} x^{2 \paren {n - j - 1} } y^{2 j}$

As an example, we examine $x^8 - y^8$:

Using the stated formula, we obtain:

which is not the same thing at all.

That they are indeed not the same can be calculated directly.

Let $x = 2, y = 1$.

We have that $2^8 - 1 = 255$.

Then we investigate what the formulae give us:

Using the wrong formula: