Common Factorizations

Beyond simple distribution.

$(a^2 - b^2) = (a + b)(a - b)$

$(a^2 + b^2) = (a + bi)(a - bi)$

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

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

An easy way to remember the difference is it goes sign → same sign → swap sign for quadratic.

$(x + y)^2 = x^2 + 2xy + y^2$

$(x - y)^2 = x^2 - 2xy + y^2$

$x^2 + bx + c = (x + \frac{b}{2})^2$ where $ \sqrt{c} = \frac{b}{2}$

This one is a bit difficult to describe, so an example would do best.

1. $x^3 - 3x^2 - x + 3$ example quadratic
2. $x^3 - 3x^2$ plus $-x + 3$ group terms with common factors
3. $x^2(x - 3)$ plus $-1(x - 3)$ reverse distribute common factors
4. $(x^2 - 1)(x - 3)$ reverse distribute the common group¹⁾