How do you factor $x^{3 a} y^{a} - y^{3 a} x^{a}$ $x^(3a) y^a - y^(3a) x^a$ ?

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How do you factor $x^{3 a} y^{a} - y^{3 a} x^{a}$ $x^(3a) y^a - y^(3a) x^a$ ?

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Answer 1 · 2015-06-03T17:13:53

Decompose the terms of $x^{3 a} y^{a} - y^{3 a} x^{a}$ $x^(3a)y^a - y^(3a)x^a$
as
$= x^{2 a} \cdot x^{a} \cdot y^{a} - x^{a} \cdot y^{a} \cdot y^{2 a}$ $= x^(2a)*x^a*y^a - x^a*y^a*y^(2a)$
$XXXX$ $color(white)("XXXX")$ now we can see an obvious common term $x^{a} y^{a}$ $x^ay^a$ =(x^ay^a)(x^(2a)-y^(2a))#

$= (x^{a} y^{a}) ({(x^{a})}^{2} - {(y^{a})}^{2})$ $=(x^ay^a)((x^a)^2 -(y^a)^2)$
$XXXX$ $color(white)("XXXX")$ the second term is the difference of squares, so
$= (x^{a} y^{a}) ((x^{a} + y^{a}) (x^{a} - y^{a})$ $=(x^ay^a)((x^a+y^a)(x^a-y^a)$

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How do you factor $x^{3 a} y^{a} - y^{3 a} x^{a}$ $x^(3a) y^a - y^(3a) x^a$ ?

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