Algebra 101 - Unit 5

Lesson 29: Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that two or more numbers or terms share. Factoring out the GCF is often the first step in simplifying or factoring polynomials.

Finding the GCF of Numbers

Steps:

1. List the factors of each number.
2. Identify the largest factor common to all numbers.

Example: GCF of 12 and 18

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6 → GCF = 6

GCF of Polynomials

Steps:

1. Find the GCF of the coefficients.
2. Find the lowest power of each variable that appears in all terms.
3. Factor out the GCF from each term.

Example 1:

Factor 6x³ + 9x²

• GCF of coefficients: 6 and 9 → 3
• GCF of variables: x³ and x² → x²
• Factor out 3x²: 6x³ + 9x² = 3x²(2x + 3)

Example 2:

Factor 12x⁴y² + 8x²y³

• GCF of coefficients: 12 and 8 → 4
• GCF of x terms: x⁴ and x² → x²
• GCF of y terms: y² and y³ → y²
• Factor out 4x²y²: 12x⁴y² + 8x²y³ = 4x²y²(3x² + 2y)

Practice Problems

1. Find the GCF of 24 and 36
2. Factor: 15x³ + 20x²
3. Factor: 18x⁴y² − 12x²y³
4. Find the GCF of 8, 12, and 20