Lesson 8.7

Greatest Common Factor (GCF)

Factoring is "un-distributing". It is the process of breaking a polynomial back into its pieces.

Introduction

We just learned how to Multiply (distribute). Now we learn how to Divide (factor). The GCF is the largest term that divides evenly into every term of the polynomial.

Past Knowledge

Lesson 1.10 (Distributive Property). . We are going backwards now.

Today's Goal

Identify and pull out the GCF from polynomials like .

Future Success

GCF is always Step 1 of EVERY factoring problem.

Key Concepts

Finding the GCF

Look for two things:

  • Number: What is the biggest number that divides all coefficients?
  • Variable: What is the smallest power of the variable shared by all?

Worked Examples

Example 1: Basic Number GCF

Basic

Factor:

1

Find GCF

What goes into 3 and 12? 3.

2

Divide

Answer

Example 2: Variables

Intermediate

Factor:

1

Smallest Power

Between and , the smaller one is .

2

Divide

Subtract exponents: .

(Don't forget the 1!)

Answer

Example 3: Coefficients & Variables

Advanced

Factor:

1

Number GCF

12 and 18. GCF is 6.

2

Variable GCF

's: is smallest.

's: is smallest.

Total GCF: .

3

Divide

First term:

Second term:

Answer

Common Pitfalls

The Disappearing 1

In , if you pull out a 5, you get . Students often write . If a term divides by itself, the answer is 1, not 0.

Not the "Greatest"

For , pulling out a 2 gives . This is wrong because 6 and 9 still share a factor (3). Keep going until nothing is left.

Real-Life Applications

Simplifying Fractions:

  • You can't cancel terms in a fraction like unless you factor first.
  • .
  • This is critical for Rational Expressions in Algebra 2.

Practice Quiz

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