Problem 13 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 12 m x-6 b x+21 a y-9 b y $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is $$6x(2m - b) + 3y(7a - 3b)$$.

Step 1: Group the terms

Group the terms with common factors: $$ (12mx - 6bx) + (21ay - 9by) $$

Step 2: Factor out the GCF from each group

Factor out the greatest common factor (GCF) from each group: $$ 6x(2m - b) + 3y(7a - 3b) $$

Step 3: Checking for a common binomial factor

From the resulting expression, it doesn't seem there is any further common binomial factor. Therefore, this is the final factored form of the polynomial: $$ 6x(2m - b) + 3y(7a - 3b) $$

Key Concepts

Grouping Method

The grouping method is a systematic technique for factoring polynomials. It's particularly useful when you have a polynomial with four or more terms. The idea is to organize or "group" the polynomial terms in a way where you can factor them conveniently.

• First, create pairs of terms that have something in common.
• Next, consider each group separately, essentially treating each as a mini-problem, where you look for a common factor.

This method is great for polynomials that do not have a straightforward common factor for all the terms, but where smaller groupings can share factors.

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is crucial in the process of factoring polynomials. The GCF is the highest number or algebraic term that divides evenly into all terms of the polynomial. It simplifies expressions and helps reduce polynomials to simpler expressions.

• Identify the smallest power of each variable that appears in all terms. This becomes part of the GCF.
• Determine the largest numerical factor common across all the terms.

Once you determine the GCF, factor it out from each group, as seen in the polynomial expression from the example above. This not only simplifies the polynomial but can also reveal further factoring opportunities.

Binomial Expressions

Binomial expressions are algebraic expressions that contain exactly two terms. These terms could be connected by addition or subtraction. Recognizing binomials is a critical step in factoring, especially with the grouping method.In the example problem, once we've grouped and factored the GCF from each set of terms, we end up with what looks like two binomial terms:

• \(2m - b\)
• \(7a - 3b\)

These expressions may or may not have a common binomial factor that further simplifies the polynomial.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables and coefficients. These variables are raised to whole-number exponents, and the structure can vary widely in complexity.The polynomial in the original problem combines four different terms with a mixture of variables:- Term 1: \(12mx\)- Term 2: \(-6bx\)- Term 3: \(21ay\)- Term 4: \(-9by\)Understanding the components of the polynomial helps identify potential simplification techniques, like using the grouping method. Moreover, recognizing each term’s variables and coefficients is crucial for subsequent steps in factoring, like finding the GCF.