Factorize $\rho(b^5 - a^5)$ to get a term with $\rho(b^3 - a^3)$

Question

I am trying to factorise $\rho(b^5 - a^5)$ so that I have a term $\rho(b^3 - a^3)$. I have a value $M = \rho(b^3 - a^3)$ that I am trying to replace $\rho$ with so when i tried to factorise it like this: $$\rho[(b^3 - a^3)(b^2 + a^2) + b^2a^3 - b^3a^2]$$ it does not work since it creates more $\rho$ values.

Is there any way to do this?

Answer 1

It's not possible in general to factor the $b^5-a^5$ term and leave a $b^3-a^3$ term. $$b^5-a^5=(b-a)(b^4+b^3a+b^2a^2+ba^3+a^4)$$ but the latter term cannot be reduced any further. Since a $b^3-a^3$ term cannot be pulled out cleanly, the task you assigned yourself is impossible.