Question

Factorize:
(5a − 7b)³ + (9c − 5a)³ + (7b − 9c)³

Answer 1

$$\begin{gathered} \mathrm{Put}\left(5a-7b\right)=x,\left(9c-5a\right)=z\mathrm{and}\left(7b-9c\right)=y. \\ \mathrm{Here}, \\ x+y+z=5a-7b+9c-5a+7b-9c=0 \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ {\left(5a-7b\right)}^3+{\left(9c-5a\right)}^3+{\left(7b-9c\right)}^3=x^3+z^3+y^3 \\ =3xzy\left[\mathrm{When}x+y+z=0,x^3+y^3+z^3=3xyz.\right] \\ =3\left(5a-7b\right)\left(9c-5a\right)\left(7b-9c\right) \end{gathered}$$