Find the following product:

$(i v)$ $(3 x - 4 y + 5 z) (9 x^{2} + 16 y^{2} + 25 z^{2} + 12 x y -$

$15 z x + 20 y z)$

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Solution

Given,

$(3 x - 4 y + 5 z) (9 x^{2} + 16 y^{2} + 25 z^{2} + 12 x y -$

$15 z x + 20 y z)$

$(3 x - 4 y + 5 z) ((3 x)^{2} + (- 4 y)^{2} + (5 z)^{2} - 3 x \times$

$(- 4 y) - (- 4 y) \times 5 z - 5 z \times 3 x$

Using Identity,

$a^{3} + b^{3} + c^{3} - 3 a b c$

$= (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - a c)$

Comparing given equation with identity we get,

$∴ (3 x)^{3} + (- 4 y)^{3} + (5 z)^{3} - 3 \times 3 x \times (- 4 y) \times 5 z =$

$(3 x - 4 y + 5 z) ((3 x)^{2} + (- 4 y)^{2} + (5 z)^{2} - 3 x \times$

$(- 4 y) - (- 4 y) \times 5 z - 5 z \times 3 x)$

$∴ (3 x - 4 y + 5 z) (9 x^{2} + 16 y^{2} + 25 z^{2} + 12 x y -$

$15 y z + 20 x z)$

$= 27 x^{3} - 64 y^{3} + 125 z^{3} + 180 x y z$

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(ii) (4x − 3y + 2z) (16x² + 9y² + 4z² + 12xy + 6yz − 8zx)

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(iv) (3x − 4y + 5z) (9x² +16y²+ 25z² + 12xy −15zx + 20yz)