Problem 51 - Algebra for College Students

Question

Factor by grouping. $$3 a x-3 b x-a y+b y$$

Step-by-Step Solution

Answer

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

Step 1: Group the Terms

First, divide the expression into two groups: \( (3ax - 3bx) + (-ay + by) \). This helps us to identify and manage like terms more effectively.

Step 2: Factor Out the Greatest Common Factor in Each Group

Look at the first group \(3ax - 3bx\). The greatest common factor (GCF) is \(3x\). Factor out \(3x\):\(3x(a - b)\).Now, look at the second group \(-ay + by\). The GCF here is \(y\). Factor out \(y\):\(-y(a - b)\).

Step 3: Simplify and Combine the Groups

Notice that both terms contain \((a - b)\) as a factor. Combine the terms to fully factor the expression:\(3x(a-b) - y(a-b)\).

Step 4: Factor Out the Common Binomial

Since \((a-b)\) is common in both terms, factor \((a-b)\) out of the expression:\((a-b)(3x-y)\).

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) plays a crucial role in simplifying expressions, especially when you need to factor by grouping. The GCF is the largest number or expression that can divide each term of an expression without leaving a remainder. Let's break down how this works.

When trying to factor an expression, first identify common elements in the terms that can be pulled out. For instance, in the expression \(3ax - 3bx\), you can spot that both terms share the number 3 and the variable \(x\). Thus, 3x is their GCF because it is the most significant factor that divides both terms evenly.

By factoring the GCF from each group, the remaining parts become easier to manage, and often, like terms will naturally align, making them even easier to factor further.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. It helps when trying to solve equations or simplify expressions, as it reveals the roots or solutions.

One common method is factoring by grouping, especially for a polynomial with four terms. Here’s how it works:

• Divide the expression into groups, managing them based on similarity or common factors.
• In each group, identify and factor out the GCF.
• Look for patterns such as common binomials, which you can factor out.

In our expression \((3ax - 3bx) + (-ay + by)\), each group was simplified by factoring out their respective GCFs. Once these are factored, look to see if there is a common binomial factor remaining in the expression, like \((a - b)\), that can be pulled out to simplify the polynomial further.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators. They are the backbone of algebra, providing a symbolic representation of relationships or formulas.

Understanding expressions involves translating these symbols into meaningful mathematical operations. For example, consider the expression \(3ax - 3bx - ay + by\). The goal is to manipulate this expression through factoring or grouping to simplify or solve equations.

When breaking down an expression for factoring by grouping:

• Identify and group similar terms to make factoring easier.
• Determine the GCF of each group and factor it out to expose commonalities.
• Finally, look to simplify the entire expression by factoring out any recurring binomials.

Manipulating expressions by finding common elements and re-grouping terms allows for more straightforward solutions and deeper understanding of the equations shaping algebraic theories and models. Learning to navigate mathematical expressions is integral in advancing further in math.