Difference of Cubes

x^3 - 64 has no pairs to group, but it has a pattern: x^3 is x cubed and 64 is 4 cubed. Any difference of two cubes factors the same way:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

With a = x and b = 4, that gives x^3 - 64 = (x - 4)(x^2 + 4x + 16). Multiply it out to see why it works: x times the long bracket is x^3 + 4x^2 + 16x, and -4 times it is -4x^2 - 16x - 64. The middle terms cancel, leaving x^3 - 64.

The quadratic factor x^2 + 4x + 16 has no real factors: its discriminant, 16 - 64, is negative, so its two roots are complex. That makes 4 the only real root of x^3 - 64. Learn the cubes 1, 8, 27, 64 and 125 so you can spot the pattern.