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

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

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Answer 1 · 2016-08-28T20:18:39

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

Explanation:

The sum of cubes identity can be written:

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

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

$1000 x^{3} + 27$ $1000x^3+27$

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

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

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

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

$= (10 x + 3) (10 x + 3 ω) (10 x + 3 ω^{2})$ $=(10x+3)(10x+3omega)(10x+3omega^2)$

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

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

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