Factor each polynomial completely. See Example 6.40ab - 16a

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial. In this case, the terms are \(40ab\) and \(-16a\).
2. Determine the GCF by finding the largest number and variable(s) that divide both terms. Here, the GCF is \(8a\).
3. Factor out the GCF from each term in the polynomial. This means you divide each term by \(8a\).
4. Write the polynomial as a product of the GCF and the remaining terms. This will look like \(8a(5b - 2)\).
5. Verify your factorization by distributing \(8a\) back through the terms in the parentheses to ensure you get the original polynomial \(40ab - 16a\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying the quadratic formula for trinomials.

Greatest Common Factor (GCF)

The greatest common factor is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, identifying the GCF allows for simplification by factoring it out, making the polynomial easier to work with. For example, in the expression 40ab - 16a, the GCF is 8a.

Difference of Squares

The difference of squares is a specific factoring technique used when a polynomial is in the form a² - b², which can be factored into (a + b)(a - b). This concept is crucial for recognizing patterns in polynomials and simplifying them effectively. While not directly applicable in this case, understanding it enhances overall factoring skills.