Question

How do you factor `x^3 + 4 = 0`?

Answer 1

`x^3+4=(x+4^(1/3))(x^2-4^(1/3)x+4^(2/3))`

`= (x+4^(1/3))(x-2^(-1/3)(1+sqrt(3)i))(x-2^(-1/3)(1-sqrt(3)i))`

Explanation:

Using the sum of cubes formula `a^3+b^3 = (a+b)(a^2-ab+b^2)` we have:

`x^3+4 = x^3+(4^(1/3))^3`

`=(x+4^(1/3))(x^2-4^(1/3)x+4^(2/3))`

If we wish to restrict ourselves to real numbers, we are done. If we allow for complex numbers, then we may continue using the quadratic formula to factor `x^2-4^(1/3)x+4^(2/3)`

`x = (4^(1/3)+-sqrt(4^(2/3)-4(1)(4^(2/3))))/2`

`=2^(-1/3)(1+-sqrt(-3))`

`=2^(-1/3)(1+-sqrt(3)i)`

Thus

`x^3+4 = (x+4^(1/3))(x-2^(-1/3)(1+sqrt(3)i))(x-2^(-1/3)(1-sqrt(3)i))`