How do you factor $x^3 - 4$ ?

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Answer 1 · 2015-04-16T16:37:10

This cannot be factored using integers.

This cannot be factored using rational numbers.

This can be factored using the irrational number $root(3)4$

$x^3 - 4 = x^3 - (root(3)4) ^3 = (x - root(3)4)(x^2 + x root(3)4 + (root(3)4)^2)$
The quadratic above cannot be factored using real numbers.

Using Imaginary numbers we can solve: $x^2 + x root(3)4 + (root(3)4)^2=0$ (Use the quadratic formula or complete the square.)

And then we can factor further:

$(x - root(3)4)(x - (root(3)4/2 + (root(3)4 sqrt3)/2 i))(x - (root(3)4/2 - (root(3)4 sqrt3)/2 i))$

How do you factor x^3 - 4x^3 - 4?