I want to factorize as $a^qb - c^q d = (b-d)(\cdots \text{a function of }a,b,c,d\cdots)$

Question

Let $2\le q\le 3$ and $a,b,c,d\ge 0$. I want to factorize the quantity as $$ a^qb - c^q d = (b-d)(\cdots \text{a function of }a,b,c,d, q\cdots). $$ Is this possible?

Answer 1

If that is possible then setting $b=d$ on the left hand side should yield $0$. Which means $a^q-c^q=0$. Since $a$ and $c$ are real and positive, this is possible when $a=c$.