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How do you factor $x^{3} + 8 y^{3}$ $x^3+8y^3$ ?

1 Answer

Dec 17, 2015

$x^{3} + 8 y^{3} = (x + 2 y) (x^{2} - 2 x y + 4 y^{2})$ $x^3+8y^3 = (x+2y)(x^2-2xy+4y^2)$

Explanation:

In general:
$XXX (a^{3} + b^{3}) = (a + b) (a^{2} - a b + b^{2})$ $color(white)("XXX")(color(red)(a)^3+color(blue)(b)^3) = (color(red)(a)+color(blue)(b))(color(red)(a)^2-color(red)(a)color(blue)(b)+color(blue)(b)^2)$

for various proofs of this see:
http://www.qc.edu.hk/math/Junior%20Secondary/sum%20of%20two%20cubes.htm

$(x^{3} + 8 y^{3}) = (x^{3} + {(2 y)}^{3})$ $(x^3+8y^3) = (color(red)(x)^3+color(blue)((2y))^3)$

Substituting
$XXX x$ $color(white)("XXX")color(red)(x)$ for $a$ $color(red)(a)$
and
$XXX 2 y$ $color(white)("XXX")color(blue)(2y)$ for $b$ $color(blue)(b)$
in the general form
$XXX$ $color(white)("XXX")$ gives the answer above.

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