Problem 69 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each sum as a product. See Example 5 \(30 a+30 b\)

Step-by-Step Solution

Answer

The expression is factored as \(30(a + b)\).

Step 1: Identify the Common Factor

Look at the expression \(30a + 30b\). Notice that 30 is a common factor in both terms, \(30a\) and \(30b\). This means we can factor out a 30 from each term.

Step 2: Apply the Distributive Property

By applying the distributive property, we factor out the common factor (30) from each term of the sum. This is done by writing the expression as a product of the common factor and a sum of the remaining terms inside parentheses: \(30(a + b)\).

Step 3: Verify the Factoring

To ensure the expression is correctly factored, distribute the 30 across the terms inside the parentheses. Re-distributing gives \(30 \times a + 30 \times b\), which simplifies to the original expression \(30a + 30b\), confirming that \(30(a + b)\) is the correct factored form.

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that involves rewriting an expression as a product of its factors. Think of it like splitting a number into its building blocks. This can make complex problems easier to solve by simplifying expressions.When you factor an expression, you aim to identify a common factor that each term shares. Once found, you "pull out" this factor, which means you divide each term by this factor and factor it out of the expression.

• For instance, in the expression \(30a + 30b\), both terms have the common factor 30.
• Factoring results in a simpler form, \(30(a + b)\), which is often more useful for solving equations or simplifying mathematical problems.

Factoring is essential because it reduces expressions and reveals deeper arithmetic properties. It's a key step in solving problems and understanding the structure of algebraic expressions.

Algebraic Expressions

Algebraic expressions use numbers, variables, and arithmetic operations (like addition and multiplication) to represent mathematical ideas. They allow you to generalize things like mathematical relationships and are fundamental in solving equations.In an algebraic expression, the variables stand in for unknown values. For example, \(a\) and \(b\) in \(30a + 30b\) could be any numbers.

• Algebraic expressions are versatile and used extensively in every aspect of algebra, often needing to be simplified or factored.
• They serve as the foundation for understanding more complicated concepts, equations, and functions in math.

Understanding algebraic expressions involves knowing how to manipulate and simplify them using properties like the distributive property. This flexibility is what makes algebra such a powerful tool in mathematics and problem-solving.

Common Factor

A common factor refers to a number or variable that divides each term in an expression without a remainder. Finding a common factor is a crucial step in simplifying algebraic expressions by factoring. In our example, the expression \(30a + 30b\) has a common factor of 30:

• The term \(30a\) can be divided by 30, leaving \(a\).
• The term \(30b\) can be divided by 30, leaving \(b\).
• Thus, the expression can be rewritten in its factored form as \(30(a + b)\).

Recognizing the common factor allows you to simplify expressions effectively. Identifying and factoring out the common factor streamlines solving and analyzing solutions in algebra.