Another major part of algebra and converting quadratics into different forms is the ability to **common factor**. Think of this as the opposite of “expanding”, which is what we did in in the previous unit. When you **factor** an expression, you’re making it more *condensed*.

Let’s start with what **factor** means. Let’s say we had the expression:

- 4(2). “Four times two” – Both the
**4**and**2**are factors. - 4(x + 3). “Four times bracket x plus 3” – This time
**4**and**(x + 3)**are factors. - 4(x + 3)(x – 9). “Four times bracket x plus 3 times bracket x minus 9” – This time
**4**,**(x + 3)**, and**(x – 9)**are factors.

Simply put, a **factor** is a number that is multiplied in order to get another number.

To find **common factors** of a polynomial, you have to look for numbers or variables that will divide out of **each** term in the expression. Take for example the expression:

- 2x² + 4. Notice that
**2**can divide out (factor out) of both terms, giving us:**2**(x² + 2). Therefore, 2 is considered the**greatest common factor**(GCF). - Similarly, if we had the binomial, 2x² + 4x, we could factor out
**2x**, giving us**2x**(x + 2). Notice that there’s one x variable in common in both terms this time. - In both examples above, the common factors happened to monomials (algebraic expression of one term). However, common factors can be binomials too – examples of these be shown in the last video link below.

To factor a polynomial:

- Find the GCF of the terms.
- Write the GCF as the first factor outside a set of brackets.
- Divide each term by the GCF, writing the result inside the brackets.

Let’s watch a video of this in action:

# Common Factor Fractions

There’s a special technique involved when common factoring expressions made up of fractions. The video below explains how such expressions are factored:

# Common Factor by Grouping

Often there is no common factor for all the terms in a polynomial, but some of the terms have a common factor. A process of **factoring by grouping** can sometimes be used with these polynomials. This process involves factoring groups of terms first, instead of factoring the entire polynomial. Here are a few examples of this in action. **Question (3)** requires that you know how to factor a **difference of squares**; this technique hasn’t been introduced yet, so you can skip it for now.

Moving forward, remember the following:Factoring a polynomial is the opposite of expanding a polynomial.