Problem 66 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each sum as a product. See Example 5 \(9 a+9 b\)

Step-by-Step Solution

Answer

The sum \( 9a + 9b \) is written as a product: \( 9(a + b) \).

Step 1: Identify the Common Factor

Look for a number that is common in both terms, 9a and 9b. The number 9 is common to both terms as a factor.

Step 2: Apply the Distributive Property

Use the distributive property which states that for any numbers or terms a, b, and c, \( a(b + c) = ab + ac \). In our case, the common factor 9 is the number that can be factored out, so we rewrite the expression as \( 9(a + b) \).

Step 3: Write the Expression as a Product

The terms 9a and 9b can be written as a product using the distributive property: \( 9(a + b) \). This means that we factor out the common factor 9 from each term in the expression.

Key Concepts

Factoring Expressions

Factoring expressions is a crucial skill in algebra that simplifies equations and makes them easier to work with. It involves breaking down an expression into simpler "factors" that when multiplied together give you the original equation. Think of factoring as the opposite of expanding – instead of distributing a multiplication, you're finding common elements to streamline the equation.
Imagine you have an expression like \( 9a + 9b \). This is actually two smaller pieces (factors) that when pulled out, simplify the problem. Factoring helps you see these factors clearly, leading to simpler computations and clearer insights into the structure of the expressions. By understanding how to factor, you can handle more complex expressions with ease. When you notice terms sharing common factors, factoring becomes a handy tool to simplify these terms and make them more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra, allowing you to construct mathematical models and solve equations.
In the expression \( 9a + 9b \), you see numbers and variables at play. Here's what makes up an algebraic expression:

• Coefficients: Numbers that multiply the variable, such as "9" in both terms.
• Variables: Symbols that represent numbers, like "a" and "b".
• Operators: Signs like "+" and "-" that show operations between terms.

Understanding these components is essential. It helps break down problems and make strategic decisions about how to manipulate the expression, whether by factoring, distributing, or combining like terms.

Common Factor

A common factor is a number or variable that appears in each term of an expression. Identifying common factors is the first step in simplifying expressions through factoring. When you recognize a common factor, it can be factored out, reducing complexity.
For instance, in the expression \( 9a + 9b \), the number 9 is a common factor of both terms. This means that 9 can be pulled out, leaving you with a simpler expression where the remaining factors inside a parenthesis are \( a + b \).
This process not only simplifies the expression but also helps to better manage equations by making them less cluttered and easier to comprehend. Identifying and using the common factor is a powerful method in algebra to unlock more straightforward resolution paths for mathematical problems.