Problem 74 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ 12 b^{2}-21 b+20 b-35 $$

Step-by-Step Solution

Answer

(4b - 7)(3b + 5)

Step 1: Group the Terms

Separate the polynomial into two groups: (12b^2 - 21b) + (20b - 35)

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Group

Identify the GCF of each group and factor it out: 3b(4b - 7) + 5(4b - 7)

Step 3: Factor Out the Common Binomial Factor

Notice that (4b - 7) is common in both terms. Factor out the common binomial: (4b - 7)(3b + 5)

Key Concepts

Grouping Terms

In polynomial expressions, factoring by grouping involves dividing the expression into smaller groups of terms. This method simplifies the process by breaking down the polynomial into more manageable parts. Let's take a closer look at how we group terms:

Consider the polynomial from the exercise: \(12b^2 - 21b + 20b - 35\).

1. We divide this polynomial into two groups: \((12b^2 - 21b)\) and \((20b - 35)\).
2. The goal is to organize the expression such that each group will have a common factor.

By grouping the terms this way, we set the foundation for factoring out common components in the next steps.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor shared by all terms in a group. Factoring out the GCF helps to simplify each group and make it easier to identify further factors.

Let's continue with our example. We have the groups \((12b^2 - 21b)\) and \((20b - 35)\):

• For the first group \(12b^2 - 21b\), we identify that the common factor is \(3b\). So, we factor out \(3b\): \(3b(4b - 7)\).
• For the second group \(20b - 35\), the common factor is \(5\). We factor out \(5\): \(5(4b - 7)\).

This step ensures each group is simpler and sets up the next step of the process where we look for a common binomial factor.

Binomial Factor

A binomial factor is an expression that consists of two terms connected by either a plus or minus sign. In the previous step, we simplified our groups to \(3b(4b - 7)\) and \(5(4b - 7)\). Now, we observe that both these groups share a common binomial factor \((4b - 7)\).

1. We factor out the common binomial factor \((4b - 7)\) from each group.
2. This gives us: \((4b - 7)(3b + 5)\).

By factoring out the common binomial factor, we have successfully factored the original polynomial fully. The expression \((4b - 7)(3b + 5)\) is the factored form of the given polynomial \(12b^2 - 21b + 20b - 35\). This method highlights how breaking down a polynomial into smaller, factorable pieces can simplify the entire factoring process.