Problem 50 - Algebra for College Students

Question

Factor by grouping. $$2 a x-b x+2 a y-b y$$

Step-by-Step Solution

Answer

The factored expression is \((x + y)(2a - b)\).

Step 1: Organize the Expression

The expression given is \(2ax - bx + 2ay - by\). First, notice that the expression can be grouped into two pairs of terms: \((2ax + 2ay)\) and \((-bx - by)\).

Step 2: Factor Out the Common Factors

In the first group \((2ax + 2ay)\), we can factor out \(2a\), resulting in \(2a(x + y)\). In the second group \((-bx - by)\), we can factor out \(-b\), resulting in \(-b(x + y)\).

Step 3: Simplify by Grouping

With both groups factored, we have: \(2a(x + y) - b(x + y)\). Notice both terms have a common factor of \((x + y)\).

Step 4: Factor out the Common Binomial

Factor out the common binomial \((x + y)\) from the expression: \((x + y)(2a - b)\).

Step 5: Write the Final Factored Expression

The fully factored form of the original expression \(2ax - bx + 2ay - by\) is \((x + y)(2a - b)\).

Key Concepts

Grouping Method

The grouping method is a strategic way to factor polynomials by organizing them into smaller groups, making the expression easier to handle. It's particularly useful when direct factoring isn't straightforward.
To apply the grouping method, first identify groups of terms that have common factors. In our example, we began with the expression:
\(2ax - bx + 2ay - by\).
We can rearrange these terms into two groups:

• \((2ax + 2ay)\)
• \((-bx - by)\)

Grouping terms helps focus on smaller parts of a more complex expression, allowing us to apply simple factoring techniques.

Common Factors

Once we have our groups, the next step is to find the common factors within them.
A common factor is a number or variable that divides each term in a set without a remainder.
For the first group, \(2ax + 2ay\), the common factor is \(2a\). This means \(2a\) can be factored out, leaving us with \(2a(x + y)\).
In the second group, \(-bx - by\), the common factor is \(-b\), so we factor it out to get \(-b(x + y)\).

• Factoring out common factors simplifies expressions and is essential for further simplifying polynomials using the grouping method.
• In both groups, finding the common factor helps us reduce more complex expressions into a binomial form.

Binomial Expression

A binomial expression is a polynomial with two terms. Recognizing binomials is crucial when factoring because it allows us to simplify even further.
In our example, after factoring out the common factors from each group, we have:

• \(2a(x + y)\)
• \(-b(x + y)\)

Both expressions feature the binomial \((x + y)\). By identifying this common binomial, we can perform another round of factoring. This is an effective technique because it reduces expressions to more basic algebraic forms.

Factored Form

The ultimate goal of the grouping method is to rewrite the expression in a simpler, factored form.
Factored form is a way of expressing a polynomial as a product of its factors, making problems easier to solve or simplify.
At the final step, the factored expression of our example is written as:
\((x + y)(2a - b)\).

• We accomplished this by recognizing \((x + y)\) as a common factor in the earlier stages of our work.
• The factored form represents the original polynomial clearly and concisely.

Understanding how to express an equation in factored form is essential for solving polynomial equations efficiently and is widely applicable in algebra.