Problem 49 - Elementary Algebra

Question

Factor \(6 x y+2 a x-3 a y-a^{2}\).

Step-by-Step Solution

Answer

Question: Factor the expression \(6xy + 2ax - 3ay - a^2\) using the distributive property in reverse (factoring by grouping). Answer: \((2x - a)(3y + a)\)

Step 1: Identify the common factors and group terms

We have the expression \(6xy + 2ax - 3ay - a^2\). We can see that the first two terms share a common factor of \(2x\), while the second two terms share a common factor of \(-a\). We can group the terms accordingly: \((6xy + 2ax) + (-3ay - a^2)\)

Step 2: Factor out the common factors from each group

Now, we will factor out the common factors from each group. For the first group, the common factor is \(2x\), and for the second group, the common factor is \(-a\): \(2x(3y + a) - a(3y + a)\)

Step 3: Factor out the common binomial

We can now see that each term has a common binomial factor of \((3y + a)\). We will factor this out to get the final factored form of the expression: \((2x - a)(3y + a)\) So, the factored form of the given expression \(6xy + 2ax - 3ay - a^2\) is \((2x - a)(3y + a)\).

Key Concepts

Common Factor

In algebra, a common factor is a number or variable that appears in all terms of an expression. By finding common factors, we can simplify algebraic expressions or equations. This is often the first step when you're trying to factor an expression. For example, consider the expression from the problem: - The first two terms, \(6xy\) and \(2ax\), share a common factor: \(2x\). - Similarly, the second two terms, \(-3ay\) and \(-a^2\), share a common factor of \(-a\).Identifying common factors makes it easier to rearrange and simplify expressions. Remember, when you factor out, you're essentially dividing each term by the common factor and bringing it outside the parentheses.

Binomial Factor

A binomial is simply a polynomial with two terms. When factoring expressions, these can sometimes appear as a common factor. Recognizing binomial factors is crucial because they can simplify solutions and reduce complex expressions. In our exercise, once we've factored out the common terms from each group, we noticed a common binomial factor: - Both groups resulted in terms involving \(3y + a\), hence it’s a common binomial factor.By factoring out this binomial, we can further simplify the expression. This shows the power of recognizing binomial factors in transforming an algebraic expression into a product of simpler expressions.

Grouping Method

The grouping method is a handy technique used in factoring when dealing with a more complex polynomial, especially those that cannot be immediately factored through simple methods. It involves rearranging terms to identify and factor out common parts step by step. Here’s the walkthrough of how we utilized the grouping method in the exercise:

• First, we looked at the four-term polynomial and decided to group them into two pairs: \((6xy + 2ax) + (-3ay - a^2)\).
• Next, for each pair, we identified and factored out their common factors. This gave us results of \(2x(3y + a)\) and \(-a(3y + a)\).
• Finally, we noticed a common binomial factor, \((3y + a)\), and factored it out completely to get \((2x - a)(3y + a)\).

This method allows you to tackle polynomials that might seem daunting at first glance by breaking them into more manageable pieces.