Problem 32 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(2 a x-6 b x-a y+3 b y\)

Step-by-Step Solution

Answer

The factored form is \((a - 3b)(2x - y)\).

Step 1: Grouping Terms

First, observe the polynomial \(2ax - 6bx - ay + 3by\) and group the terms that have common factors. We can group it as \((2ax - 6bx) + (-ay + 3by)\).

Step 2: Factor Out Common Factors in Each Group

In the first group \((2ax - 6bx)\), both terms have a common factor of \(2x\). Factoring \(2x\) out gives us \(2x(a - 3b)\). In the second group \((-ay + 3by)\), both terms have a common factor of \(y\). Factoring \(y\) out gives us \(-y(a - 3b)\).

Step 3: Factor the Common Binomial

Now the expression is \(2x(a - 3b) - y(a - 3b)\). Notice \((a - 3b)\) is a common factor in both terms. Factor \((a - 3b)\) out to get \((a - 3b)(2x - y)\).

Key Concepts

Grouping Method

The grouping method is a systematic approach for factoring polynomials, especially when direct factoring is not straightforward. Here's how it works:

• Identify terms within the polynomial that can be "grouped" together based on commonality.
• Each group must have a common factor that can be later factored out.

In the given exercise, the polynomial is arranged as two separate groups:
deconstructed into from the expression \(2ax - 6bx - ay + 3by\) which we grouped into \((2ax - 6bx) + (-ay + 3by)\). Grouping allows easier identification and factorization of common factors. By successfully grouping terms, you set the stage for efficient factorization in subsequent steps.

Common Factors

Common factors are shared by terms, which simplifies expressions through factorization.

• To identify a common factor, compare each term in your grouped polynomial.
• Pull out the greatest common factor from each group, which simplifies the terms.

For example, from the first group \((2ax - 6bx)\), the common factor is \(2x\). Factoring \(2x\) yields \(2x(a - 3b)\). The second group \((-ay + 3by)\) has a common factor of \(y\), resulting in \(-y(a - 3b)\).
Understanding and finding common factors is critical as it reduces polynomial complexity, making them simpler to handle.

Binomial Factorization

Once individual groups are simplified, binomial factorization comes into play. This involves recognizing a common binomial factor across terms.

• Look for a binomial factor that is repeated in the expression.
• Factor out this binomial factor to further simplify.

In this exercise, the factor \((a - 3b)\) is identified in both terms, resulting in the expression \(2x(a - 3b) - y(a - 3b)\). Act on this shared factor and factorize it out, the equation becomes \((a - 3b)(2x - y)\). Binomial factorization streamlines the expression, highlighting its core structure.