Factoring

Factoring or factorising in British English is the process of finding the factors an algebraic term, or writing a term as a multiple of more simple terms. It is the opposite of expanding. A simple example would be:

$3x + 9 = 3(x+3)$
With the first term being the expanded version and the second factored. A more complex example would be:

$2x^2 + 5x + 3 = (2x + 3)(x + 1)$

Common factoring patterns

Binomials

Binomials tend to appear by themselves as a square:

$\begin{align} a^2+2ab+b^2 = (a+b)^2\\ a^2-2ab+b^2 = (a-b)^2\end{align}$

If raised to higher powers, they continue according to the pattern shown in the Binomial coefficient:

$(a+b)^3 = a^3 + 3a^2b + 3 b^2a + b^3$

Quadratic formula

The Quadratic formula determines roots, which can be factored from the polynomial.

$f(x) = ax^2 + bx + c = 0,\quad a \ne 0$

$\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Difference of squares

One common factoring problem is called a difference of squares. This involves a squared factor minus another squared factor. This type of equation can be written as follows:

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

An example of this using real numbers would be:

$4x^2 - 9y^2 = (2x - 3y)(2x + 3y)$

Sum of Squares

Another common factoring problem is the sum of squares. similar to difference of squares, its results are quite different.

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

Involving the definition of complex numbers, the sum of squares can be derived from the difference of squares formula.

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

$a^2 + (-1 * b^2) = (a + b)(a - b)$

$a^2 + (\sqrt{-1}b)^2 = (a + b)(a - b)$

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

$c = ib$

$a^2 + c^2 = (a + \frac c{i})(a - \frac c{i})$

$a^2 + c^2 = (a + \frac {ci}{i^2})(a - \frac {ci}{i^2})$

$a^2 + c^2 = (a - ci)(a + ci)$

An example would be

$4x^2 + 9y^2 = (2x + 3yi)(2x - 3yi)$