Question
Factorize the following expressions:
$8x^3 + 27y^3 - 216z^3 + 108xyz$
$8x^3 + 27y^3 - 216z^3 + 108xyz$
✓
Answer
$8 x^3+27 y^3-216 z^3+108 x y z$
$=(2 x)^3+(3 y)^3+(-6 z)^3-3(2 x)(3 y)(-6 z)$
$=(2 x+3 y-6 z)\left((2 x)^2+(3 y)^2+(-6 z)^2-2 x \times 3 y-3 y(-6 z)-(-6 z) 2 x\right)$
$ =(2 x+3 y-6 z)\left(4 x^2+9 y^2+36 z^2-6 x y+18 y z+12 z x\right)$
$\therefore 8x^3 + 27y^3 - 216z^3 + 108xyz$
$=(2 x+3 y-6 z)\left(4 x^2+9 y^2+36 z^2-6 x y+18 y z+12 z x\right)$
$=(2 x)^3+(3 y)^3+(-6 z)^3-3(2 x)(3 y)(-6 z)$
$=(2 x+3 y-6 z)\left((2 x)^2+(3 y)^2+(-6 z)^2-2 x \times 3 y-3 y(-6 z)-(-6 z) 2 x\right)$
$ =(2 x+3 y-6 z)\left(4 x^2+9 y^2+36 z^2-6 x y+18 y z+12 z x\right)$
$\therefore 8x^3 + 27y^3 - 216z^3 + 108xyz$
$=(2 x+3 y-6 z)\left(4 x^2+9 y^2+36 z^2-6 x y+18 y z+12 z x\right)$
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