Sum of Cubes

A sum of two cubes factors too, with the signs moved around:

$${a}^{3}+{b}^{3} = \left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)$$

Compare it with the difference: the short bracket has the same sign as the cubes, the middle of the long bracket has the opposite sign, and the last term is always plus. The cubes can carry a number. In $27{x}^{3}+8$, $27{x}^{3}$ is ${\left(3x\right)}^{3}$ and 8 is 2 cubed, so a = 3x and b = 2. Then a squared is $9{x}^{2}$ and ab is 6x:

$$27{x}^{3}+8 = \left(3x+2\right)\left(9{x}^{2}-6x+4\right)$$

Unlike a sum of squares, a sum of cubes always has a real linear factor: ${a}^{3}+{b}^{3}$ is zero when a = -b, so $a+b$ is a factor. Multiply out to check: $-18{x}^{2}$ and $18{x}^{2}$ cancel, and so do -12x and 12x.