Problem 88 - Elementary Algebra

Question

Factor completely. $$ (a+3) 3+(b-3) 3 $$

Step-by-Step Solution

Answer

3(a + b)

Step 1: Identify Common Factor

Review the expression \((a+3)3 + (b-3)3\). Notice that both terms contain a common factor of 3. This means that the expression can be factored by taking 3 as a common factor.

Step 2: Factor Out the Common Factor

Factor out 3 from the expression: \(3(a+3) + 3(b-3)\). This simplifies to \(3[(a+3) + (b-3)]\).

Step 3: Simplify the Expression Inside the Brackets

Simplify the expression inside the brackets: \((a+3) + (b-3) = a + 3 + b - 3\). This simplifies further to \(a + b\).

Step 4: Write the Final Factored Expression

Substitute the simplified expression \(a + b\) back into the factorized form to get the final result: \(3(a + b)\).

Key Concepts

Understanding Common Factor

In algebra, identifying a common factor is an essential skill that helps simplify expressions and solve equations. A common factor is a number or expression that divides each term within an algebraic expression without leaving a remainder. In the expression

• \((a+3)3 + (b-3)3\), the number 3 appears in both terms as a factor, making it the common factor.

By recognizing a common factor in algebraic expressions, you can simplify calculations and find easier ways to manipulate the algebraic equations. Extracting the common factor helps to combine and simplify terms. Therefore, it often serves as the first step in the process of factoring complex algebraic expressions.Identifying the common factor plays a critical role not just for simplification, but also for making the expression more understandable and manageable. It's like peering through the algebraic clutter and gaining a clear path toward a streamlined solution. Here, focusing on the common factor 3 sets the stage for further simplification.

Decoding Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators that are grouped together. They form the building blocks of algebraic equations and can represent real-world quantities or abstract mathematics.Consider the given exercise:

• \((a+3)3 + (b-3)3\). This expression includes two separate terms, each with its own elements contained within parentheses, and the multiplier 3, which is a number."

Algebraic expressions like this one can initially seem daunting due to the mix of symbols and variable terms. However, by breaking them down into their components, identifying common factors, and strategically simplifying them, even complex expressions become more approachable.Notably, expressions can often be rewritten with fewer terms through processes like factoring. Understanding the components and structure of an algebraic expression is key to effectively managing and simplifying them as encountered in both theoretical and practical mathematical problems. By working systematically through the expressions, you build a strong foundation applicable across various math contexts.

The Simplification Process Made Clear

Simplifying an algebraic expression means reducing it to its most basic form while retaining its original value. This process involves several steps, each crucial for achieving a concise outcome. Let's explore:First, in the provided expression:

• \((a+3)3 + (b-3)3\), we began by factoring out the common factor (3), leading us to: \(3[(a+3) + (b-3)]\).

This eliminates the repeated multiplication and sets the stage for further simplification inside the brackets. By removing any parentheses and performing simple arithmetic:

• \((a+3) + (b-3)\) is simplified to \(a + b\).

Finally, substitute this back, yielding the complete simplified expression: \(3(a + b)\).Simplification not only aids in revealing the essential components of an expression, but also helps in solving equations more quickly and accurately. The process removes complexities by focusing on common elements and performing basic arithmetic operations, ultimately ensuring that the expression is as straightforward as possible. This clarity makes further mathematical manipulation or application more intuitive, simplifying both understanding and use.