How do you factor $8x^3y^6 + 27$ ?

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Answer 1 · 2015-04-22T13:01:44

$a^3+b^3 = (a+b)(a^2-ab+b^2)$
(this is either something you already know or you can derive it by synthetic division/multiplication)

$8x^3y^6+27 = (2xy^2)^3+3^3$

So this can be factored as
$(2xy^2+3)((2xy^2)^2-(2xy^2)(3)+3^2)$
or
$(2xy^2+3)(4x^2y^4-6xy^2+9)$

You might be tempted to try to factor the second part of this but if you consider the formula for roots of a quadratic:
$(-b+-sqrt(b^2-4ac))/(2a)$

we have the square root of a negative value; so there are no Real factors available.

How do you factor 8x^3y^6 + 278x^3y^6 + 27?