Question
Factorise: $(2 x-3 y)^3+(3 y-4 z)^3+(4 z-2 x)^3$
✓
Answer
$\text { Let } a=2 x-3 y, b=3 y-4 z, c=4 z-2 x$
$\text { then } a+b+c=2 x-3 y+3 y-4 z+4 z-2 x=0$
$\therefore a^3+b^3+c^3=3 a b c$
$(2 x-3 y)^3+(3 y-4 z)^3+(4 z-2 x)^3$
$=3(2 x-3 y)(3 y-4 z)(4 z-2 x)$
$=3(2 x-3 y)(3 y-4 z) \times 2(2 z-x)$
$=6(2 x-3 y)(3 y-4 z)(2 z-x)$
$\text { then } a+b+c=2 x-3 y+3 y-4 z+4 z-2 x=0$
$\therefore a^3+b^3+c^3=3 a b c$
$(2 x-3 y)^3+(3 y-4 z)^3+(4 z-2 x)^3$
$=3(2 x-3 y)(3 y-4 z)(4 z-2 x)$
$=3(2 x-3 y)(3 y-4 z) \times 2(2 z-x)$
$=6(2 x-3 y)(3 y-4 z)(2 z-x)$
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