$\displaystyle a^{3}+b^{3}=( \sqrt[3]{a^{3}}+ \sqrt[3]{b^{3}}) (a^{3-1} -\sqrt[3]{a^3b^3} +b^{3-1})$

And just switch the sign after $\displaystyle a^3-1$ for$\displaystyle a^3-b^3$

My question:

I have read that the last two signs are constant (never changing) Unless the initial sign is different. I.e.

$\displaystyle a-b = +,+$

$\displaystyle a+b = -,+ $

now take this example:

$\displaystyle 8x^3-1$

when worked out it equals:

$\displaystyle (\sqrt [3] {8x^3} + \sqrt [3] {-1})(8x^{3-1}+(-1)2x+(-1^{3-1})$

$\displaystyle (2x-1)(4x^2-2x-1)$

The end signs are clearly not $\displaystyle +,+$

Am I missing something?