Factor Polynomials by Using GCF

Algebra-1

1. Fundamental Concepts

Definition: Factoring polynomials involves expressing a polynomial as a product of its factors.
Greatest Common Factor (GCF): The largest expression that divides evenly into each term of the polynomial.
Factorization Process: Identifying and factoring out the GCF from each term in the polynomial.

2. Key Concepts

Basic Rule: $$(ax + bx) = x(a + b)$$

Degree Preservation: The degree of the polynomial remains unchanged after factoring out the GCF.

Application: Factoring is used to simplify expressions, solve equations, and analyze functions.

3. Examples

Example 1 (Basic)

Problem: Factor the polynomial $$(6x^2 + 9x)$$

Step-by-Step Solution:

Identify the GCF: $$\text{GCF}(6x^2, 9x) = 3x$$
Factor out the GCF: $$(6x^2 + 9x) = 3x(2x + 3)$$

Validation: Substitute $x=1$ → Original: $6+9=15$ ; Simplified: $3 \cdot (2+3)=15$ ✓

Example 2 (Intermediate)

Problem: Factor the polynomial $$(12y^3 - 8y^2 + 4y)$$

Step-by-Step Solution:

Identify the GCF: $$\text{GCF}(12y^3, -8y^2, 4y) = 4y$$
Factor out the GCF: $$(12y^3 - 8y^2 + 4y) = 4y(3y^2 - 2y + 1)$$

Validation: Substitute $y=1$ → Original: $12-8+4=8$ ; Simplified: $4 \cdot (3-2+1)=8$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to highlight common factors in each term.
Error-Proofing: Double-check by distributing the GCF back through the parentheses to ensure the original polynomial is recovered.
Concept Reinforcement: Practice with a variety of polynomials to reinforce understanding of different factorization scenarios.