How do you factor $1000x^3+27$ ?

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Answer 1 · 2016-08-28T20:18:39

$1000x^3+27=(10x+3)(100x^2-30x+9)$

The sum of cubes identity can be written:

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

Use this with $a=10x$ and $b=3$ as follows:

$1000x^3+27$

$=(10x)^3+3^3$

$=(10x+3)((10x)^2-(10x)(3)+3^2)$

$=(10x+3)(100x^2-30x+9)$

This is as far as we can go with Real coefficients. If we allow Complex coefficients then it can be factored further as:

$=(10x+3)(10x+3omega)(10x+3omega^2)$

where $omega = -1/2+sqrt(3)/2i$ is the primitive Complex cube root of $1$ .

How do you factor 1000x^3+271000x^3+27?