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. For the terms \$40ab\( and \)16a$, find the largest number and variable factors that divide both terms.
2. Determine the GCF of the coefficients: 40 and 16. Also, determine the common variable factors. In this case, both terms have the variable \(a\).
3. Write the polynomial as a product of the GCF and the remaining terms inside parentheses. This means expressing \$40ab - 16a$ as \(\text{GCF} \times (\text{remaining terms})\).
4. Divide each term of the polynomial by the GCF to find the terms inside the parentheses. For example, divide \$40ab\( by the GCF and \)16a$ by the GCF separately.
5. Write the fully factored form as \(\text{GCF} \times (\text{result of division for first term} - \text{result of division for second term})\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides two or more terms without leaving a remainder. Factoring out the GCF simplifies polynomials by extracting common numerical coefficients and variables, making the expression easier to work with.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process often starts by identifying and factoring out the GCF, which reduces the polynomial to a simpler form that can be further analyzed or solved.

Properties of Variables and Exponents

Understanding how variables and their exponents behave during multiplication and division is essential. When factoring, common variables with the smallest exponents are factored out, ensuring the remaining terms are simplified correctly without changing the polynomial's value.