Problem 302 - Elementary Algebra

Question

Factor completely. \(12 a b-6 a+10 b-5\)

Step-by-Step Solution

Answer

The completely factored form is (2b - 1)(6a + 5).

Step 1: Group the terms

Group the terms in pairs to make factoring easier. Group the first two terms together and the last two terms together: \[ (12ab - 6a) + (10b - 5) \]

Step 2: Factor out the greatest common factor (GCF) from each group

Identify and factor out the greatest common factor from each group: \[ 6a(2b - 1) + 5(2b - 1) \]

Step 3: Factor out the common binomial factor

Notice that both terms now contain a common binomial factor (2b - 1). Factor this out: \[ (2b - 1)(6a + 5) \]

Key Concepts

Grouping Terms

Grouping terms is the first step in factoring a polynomial, particularly when it involves more than two terms. By grouping terms, we re-organize them into pairs or smaller sets to make the factoring process more manageable. In our exercise, we start with the polynomial: \( 12ab - 6a + 10b - 5 \) To group the terms, you can look for pairs that can be factored more easily. In this case, we can pair the first two terms and the last two terms: \( (12ab - 6a) + (10b - 5) \) This grouping helps us to see common factors within each pair that we can factor out in the next step.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. Factoring out the GCF simplifies the expressions within each group. Once we have grouped the terms, we find the GCF for each group. Looking at the groups \( (12ab - 6a) \) and \( (10b - 5) \), we identify the GCF for each:

• For \( 12ab - 6a \), the GCF is \( 6a \).
• For \( 10b - 5 \), the GCF is \( 5 \).

We factor out these GCFs: \( 6a(2b - 1) + 5(2b - 1) \) Factoring out the GCF simplifies each term and brings us to the next step of factoring completely.

Binomial Factor

A binomial factor is an expression that contains two terms. After factoring out the GCF from each group, we notice that both factored groups share a common binomial factor. In our example, \( (2b - 1) \) is the common binomial factor for both groups: \( 6a(2b - 1) + 5(2b - 1) \) When a common binomial factor is present, we can factor it out completely to combine the expressions. This is what we do in the final step: \( (2b - 1)(6a + 5) \) This results in the completely factored form of the polynomial. Understanding the role of common binomial factors helps in simplifying and solving polynomials efficiently.