Question 35
Find the following product:
$(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$

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Solution

$(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$
$2 x (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z) - y (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z) + 3 z (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$
$= 8 x^{3} + 2 x y^{2} + 18 x z^{2} + 4 x^{2} y + 6 x y z - 12 x^{2} z - 4 x^{2} y - y^{3} - 9 y z^{2} - 2 x y^{2}$
$- 3 y^{2} z + 6 x y z + 12 x^{2} z + 3 y^{2} z + 27 z^{3} + 6 x y z + 9 y z^{2} - 18 x z^{2}$
$= 8 x^{3} + (2 x y^{2} - 2 x y^{2}) + (18 x z^{2} - 18 x z^{2}) + (4 x^{2} y - 4 x^{2} y) + (6 x y z + 6 x y z + 6 x y z)$
$+ (- 12 x^{2} z + 12 x^{2} z) - y^{3} + (- 9 y z^{2} + 9 y z^{2}) + (- 3 y^{2} z + 3 y^{2} z) + 27 z^{3}$
$= 8 x^{3} + 18 x y z - y^{3} + 27 z^{3}$
$= 8 x^{3} - y^{3} + 27 z^{3} + 18 x y z$

Alternate method:

$(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$
$= (2 x - y + 3 z) (2 x)^{2} + (- y)^{2} + (3 z)^{2} - (2 x) (- y) - (- y) (3 z) - (2 x) (3 z)$
$= (2 x)^{3} + (- y)^{3} + (3 z)^{3} - 3 (2 x) (- y) (3 z)$
$[U s i n g t h e i d e n t i t y, (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = a^{3} + b^{3} + c^{3} - 3 a b c]$
$= 8 x^{3} - y^{3} + 27 z^{3} + 18 x y z$

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Similar questions

(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)

(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)

2 x y - 4 x + y = 17, 3 y z + y - 6 z = 52, 6 x z + 3 z + 2 x = 29.

(x + 2 y + 3 z)^{2}

x^{2} + y^{2} - z^{2} = 21

3 x z + 3 y z - 2 x y = 18

x + y - z = 5

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