Factoring and the Greatest Common Factor

Introduction to Factoring

Factoring is a fundamental process in algebra that involves rewriting an expression as a product of its factors. This technique is essential for simplifying expressions, solving equations, and understanding polynomial structure.

• Factoring is the process of expressing a number or algebraic expression as a product of its factors.

• Factoring is used to simplify expressions and solve equations efficiently.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor shared by all terms in a list. It can refer to integers or variables in algebraic expressions.

• GCF of Integers: The largest integer that divides each number in the list without a remainder.

• GCF of Variables: The variable raised to the lowest power that appears in all terms.

Examples: Finding the GCF

• Example 1a: Find the GCF of 30, 75, and 135.

  • Prime factorization:

    • 30 = 2 × 3 × 5

    • 75 = 3 × 5 × 5

    • 135 = 3 × 3 × 3 × 5

  • Common factors: 3 and 5

  • GCF = 15

• Example 1b: Find the GCF of , , .

  • For variable p: lowest power is 1 (from )

  • For variable q: lowest power is 1 (from )

  • GCF =

• Example 1c: Find the GCF of , .

  • Numerical GCF: GCF of 21 and 14 is 7

  • Variable GCF: lowest power of x is 1

  • GCF =

• Example 1d: Find the GCF of , , .

  • Numerical GCF: GCF of 15, 5, and 20 is 5

  • Variable GCF: lowest power of y is 2

  • GCF =

Factoring Out the Greatest Common Factor

Factoring out the GCF is a method used to simplify expressions by removing the largest common factor from all terms. This process is often the first step in factoring polynomials.

• Step 1: Find the GCF of all terms in the expression.

• Step 2: Divide each term by the GCF.

• Step 3: Write the expression as the product of the GCF and the resulting simplified expression.

Examples: Factoring Out the GCF

• Example 2a: Factor

  • GCF: 4x

  • Divide each term:

  • Factored form:

• Example 2b: Factor

  • GCF: 2b

  • Divide each term:

  • Factored form:

• Example 2c: Factor

  • GCF: 2y^2

  • Divide each term:

  • Factored form:

• Example 2d: Factor

  • GCF: 3n

  • Divide each term:

  • Factored form:

Summary Table: GCF Examples

Key Takeaways

• The GCF is the largest factor common to all terms, including both numbers and variables.

• Factoring out the GCF simplifies expressions and is often the first step in solving polynomial equations.

• Always check for a GCF before attempting more advanced factoring techniques.