What Is Factorization?

In algebra, you've already learned how to expand expressions — for example, turning $3 (x + 2)$ into $3 x + 6$ . Factorization is the reverse process. It means rewriting an expression as a product of its factors.

Factorization

The process of writing an algebraic expression as a product of two or more factors. It is the reverse of expanding brackets.

Factor

A number, variable, or expression that divides evenly into another expression with no remainder.

For example:

$3 x + 6$ can be factorized as $3 (x + 2)$
The factors are $3$ and $(x + 2)$

Here's a concrete example before we go further:

$8 x + 12 = 4 (2 x + 3)$

We've rewritten the sum $8 x + 12$ as the product $4 \times (2 x + 3)$ . That's factorization!

Analogy

Think of it like this: expanding is "unpacking a gift box," and factorizing is "packing everything neatly back into the box." Just as you might unpack a box to find 3 sets of (1 book + 2 pens), you can pack $3 x + 6$ back into the box $3 (x + 2)$ .

What Is a Common Factor?

Before you can factorize, you need to identify the common factor — something that divides evenly into every term of the expression.

Common Factor

A number, variable, or combination of both that is a factor of every term in an expression.

Let's look at the expression $6 x + 9$ :

The coefficient of the first term is $6$ . Its positive integer factors are: $1, 2, 3, 6$ .
The coefficient of the second term is $9$ . Its positive integer factors are: $1, 3, 9$ .
Common factors of the coefficients: $1$ and $3$ .
The highest common factor (HCF) of the coefficients is $3$ .
Since the second term $9$ has no variable, there is no variable common factor.
So the overall HCF is $3$ , giving: $6 x + 9 = 3 (2 x + 3)$ .

We always want to take out the highest common factor to fully factorize the expression.

Exam Tip

To find the HCF of the numerical parts, list the positive integer factors of each coefficient and pick the largest one they share. For variables, choose the lowest power of each variable that appears in every term.

Analogy

Imagine you have 3 apples and 6 oranges, and you want to split them equally into bags. You could put them into 3 bags, each containing 1 apple and 2 oranges. That's exactly what factorization does — it finds the biggest "bag" (the common factor) that everything fits into equally.

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