How do you factor $125x^3+144$ ?

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Answer 1 · 2015-12-13T00:51:37

$125x^3 + 144 =(5x + 144^(1/3))(25x^2 - 5(144^(1/3))x + 144^(2/3))$

The sum of two cubes can be factored as
$a^3 + b^3 = (a+b)(a^2 -ab + b^2)$

While unfortunately 144 is not a perfect cube, it still a cube of some number. Specifically it is the cube of $144^(1/3)$

Applying this, we have

$125x^3 + 144 = (5x)^3 + (144^(1/3))^3$

$=(5x + 144^(1/3))(25x^2 - 5(144^(1/3))x + 144^(2/3))$

With a sum of difference of cubes we always get a single real root, meaning we cannot factor any further. So our final factorization is

$125x^3 + 144 =(5x + 144^(1/3))(25x^2 - 5(144^(1/3))x + 144^(2/3))$

How do you factor 125x^3+144125x^3+144?