Question
Factorise: $(5a - 7b)^3 + (7b - 9c)^3 + (9c - 5a)^3$
✓
Answer
Put $(5a - 7b) = x, (7b - 9c) = y, (9c - 5a) = z.$
Here, $x + y + z = 5a - 7b + 9c - 5a + 7b - 9c = 0$ Thus,
We have: $(5a - 7b)^3 + (9c - 5a)^3 + (7b - 9c)^3 = x^3 + z^3 + y^3 = 3xyz$
[When $x + y + z = 0, x^3 + y^3 + z^3 = 3xyz] $
$= 3(5a - 7b)(9c - 5a)(7b - 9c)$
Here, $x + y + z = 5a - 7b + 9c - 5a + 7b - 9c = 0$ Thus,
We have: $(5a - 7b)^3 + (9c - 5a)^3 + (7b - 9c)^3 = x^3 + z^3 + y^3 = 3xyz$
[When $x + y + z = 0, x^3 + y^3 + z^3 = 3xyz] $
$= 3(5a - 7b)(9c - 5a)(7b - 9c)$
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