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))#