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.