Factoring: Extracting Common Factors

Article Contents

What is Factoring
Simple Factoring
Factoring Multiple Terms
Factoring with Parentheses
Common Mistakes
Practice Examples
Interactive Exercises

1. What is Factoring {#what-is-factoring}

Factoring is the reverse of expanding parentheses. We "pull out" a common factor in front of parentheses.

ab + a c = a (b + c)

Why is it Useful?

Simplifies expressions
Helps solve equations
Reveals structure of expressions

2. Simple Factoring {#simple-factoring}

Numerical Common Factor

3 x + 6 = 3 (x + 2)

Verification: $3 (x + 2) = 3 x + 6$ ✓

Variable Common Factor

x^{2} + x = x (x + 1)

Because $x$ appears in both terms.

Negative Common Factor

4 x - 8 = 4 (x - 2)

3. Factoring Multiple Terms {#factoring-multiple-terms}

Two Terms

6 x + 9 y = 3 (2 x + 3 y)

Common divisor is $3$ .

Three Terms

x^{2} + 2 x + 1 = (x + 1) (x + 1) = (x + 1)^{2}

Greatest Common Divisor (GCD)

For numbers $12$ and $18$ , GCD = $6$ :

12 x + 18 = 6 (2 x + 3)

4. Factoring with Parentheses {#factoring-with-parentheses}

More Complex Case

3 (x + 2) + y (x + 2)

Common parentheses $(x + 2)$ :

(x + 2) (3 + y)

Negative Term Variant

2 (x - 3) - 5 (x - 3)

(x - 3) (2 - 5) = (x - 3) (- 3) = - 3 (x - 3)

5. Common Mistakes {#common-mistakes}

❌ Mistake 1: Incomplete Factoring

4 x + 8 = 2 (2 x + 4) = 4 (x + 2) (CORRECT)

4 x + 8 = 2 (2 x + 4)

is also correct, just not fully factored.

❌ Mistake 2: Wrong Common Factor

6 x + 9 \neq = 3 (2 x + 3) (WRONG!)

Verification: $3 (2 x + 3) = 6 x + 9$ ✓

6. Practice Examples {#practice-examples}

Example 1: Numerical Expression

15 + 25 x = 5 (3 + 5 x)

Example 2: Variable

x^{2} - 4 x = x (x - 4)

Example 3: Both Variants

6 x^{2} + 9 x = 3 x (2 x + 3)

7. Interactive Exercises {#interactive-exercises}

Practice factoring:

Summary

Step	Example
Find common factor	$a$ in $a x + a y$
Factor out front	$a x + a y = a (x + y)$
Verify by expanding	$a (x + y) = a x + a y$ ✓

Introduction to Algebra – basic concepts
Expanding – reverse process
Difference of Squares – special case

Factoring Out Common Terms