How do you factor $x^6y^3 + y^9$ ?

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Answer 1 · 2017-02-25T15:32:21

$x^6y^3+y^9 = y^3(x^2+y^2)(x^2-sqrt(3)xy+y^2)(x^2+sqrt(3)xy+y^2)$

The sum of cubes identity can be written:

$a^3+b^3 = (a+b)(a^2-ab+b^2)$

So we find:

$x^6y^3+y^9 = y^3(x^6+y^6)$

$color(white)(x^6y^3+y^9) = y^3((x^2)^3+(y^2)^3)$

$color(white)(x^6y^3+y^9) = y^3(x^2+y^2)(x^4-x^2y^2+y^4)$

Then note that:

$(x^2-axy+y^2)(x^2+axy+y^2) = x^4+(2-a^2)x^2y^2+y^4$

So putting $a=sqrt(3)$ we find:

$x^4-x^2y^2+y^4 = (x^2-sqrt(3)xy+y^2)(x^2+sqrt(3)xy+y^2)$

Putting it all together:

$x^6y^3+y^9 = y^3(x^2+y^2)(x^2-sqrt(3)xy+y^2)(x^2+sqrt(3)xy+y^2)$

How do you factor x^6y^3 + y^9x^6y^3 + y^9?