**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 - b)^2 $

An example would be

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