11.1. Common Factors

As usual, we must start with definitions of the terms we will be using in the following sections.

11.1.1. Factors of an Expression

The factors of an expression are those quantities whose product is the original expression.

Example 1:

The factors of x2 − 9 are x + 3 and x − 3, because

Many expressions have no factors other than 1 and themselves. Such expressions are called prime.

Factoring is the process of finding the factors of an expression. It is the reverse of finding the product of two or more quantities.


x(x + 4) = x2 + 4x


x2 + 4x = x(x + 4)

We usually factor an expression by recognizing the form of that expression. In the first type of factoring we will cover, we look for common factors.

11.1.2. Common Factors

▪ Exploration:

Try this. Expand this expression by multiplying out.

a(b + c + d)

Now how would you return the expression you just got back to its original form? Can you state your findings as a general rule?

If each term of an expression contains the same quantity (called the common factor), the quantity may be factored out.


This is nothing but the distributive law that we studied earlier.

Example 2:

In the expression

x3 − 3x

each term contains an x as a common factor. So we write

x3 − 3x = x (x2 − 3)

Most of the factoring ...

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