Problem 79 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ r s-r u+8 s w-8 u w $$

Step-by-Step Solution

Answer

The factored form is \((s-u)(r+8w)\).

Step 1: Group the Terms

First, group the terms of the expression in pairs that may factor easily: \( (r s - r u) + (8 s w - 8 u w) \).

Step 2: Factor Out Common Factors in Each Group

Next, factor out the greatest common factor from each group. In the first group \( (r s - r u) \), factor out \( r \): \( r(s-u) \). In the second group \( (8 s w - 8 u w) \), factor out \( 8w \): \( 8w(s-u) \).

Step 3: Apply the Distributive Property

Notice that both groups contain the common factor \( (s-u) \). Factor \( (s-u) \) out of the entire expression: \((s-u)(r+8w)\). The expression is now fully factored.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operators (like addition, subtraction, multiplication, and division). These expressions can represent a multitude of real-world problems and scenarios, making them a fundamental part of algebra. In the expression \( rs - ru + 8sw - 8uw \), we find different terms all strung together by operation signs.
Understanding each part is crucial:

• **Terms**: Each part of the expression separated by a plus or minus sign, such as \( rs \) or \( -ru \).
• **Coefficients**: The numbers multiplying the variables, like \( r \), which is the coefficient of the term \( rs \).
• **Variables**: Represent unknown quantities, such as \( r, s, u, \) and \( w \).

Recognizing how these components interact is the first step in mastering algebra. Simplifying expressions like this involves grouping terms and finding common factors, a basic process in algebraic manipulation.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms. Identifying the GCF is a key step in simplifying expressions and is particularly useful in factoring by grouping. Let's break down the term \( rs - ru \) from our example to find the GCF.

• **Step 1**: Identify common factors in the terms. For \( rs \) and \( ru \), the common factor is \( r \).
• **Step 2**: Factor \( r \) out of both terms, resulting in \( r(s-u) \).

For the terms \( 8sw - 8uw \), the process is similar:

• **Step 1**: The common factors are \( 8 \) and \( w \).
• **Step 2**: Factor \( 8w \) out, yielding \( 8w(s-u) \).

By factoring out these greatest common factors, the expression simplifies and makes it easier to apply further algebraic strategies, like the distributive property.

Distributive Property

The distributive property is a fundamental algebraic principle used to multiply a single term across terms inside parentheses. It's written as \( a(b + c) = ab + ac \). This property is integral when factoring algebraic expressions. In the problem \( rs - ru + 8sw - 8uw \), after grouping and factoring out the common terms, we get:

• First group factorization: \( r(s-u) \)
• Second group factorization: \( 8w(s-u) \)

Both groups share the common factor \( (s-u) \). Now, re-apply the distributive property:

• Extract \( (s-u) \) from both terms, resulting in \( (s-u)(r+8w) \).

This step shows how the distributive property not just distributes terms but also allows factoring by regrouping. This is a powerful strategy in algebra to simplify and solve equations efficiently.