Problem 64 - Algebra A Combined Function

Question

Use the distributive property to write each sum as a product. See Examples 13 and 14. $$ 14 \cdot z+14 \cdot 5 $$

Step-by-Step Solution

Answer

The product is \( 14 \cdot (z + 5) \).

Step 1: Identify Common Factor

Look at both terms of the sum, \( 14 \cdot z \) and \( 14 \cdot 5 \), and identify the common factor. In this case, the common factor is \( 14 \).

Step 2: Apply the Distributive Property

Factor out the common factor from the expression. The distributive property tells us that \( a \cdot b + a \cdot c = a \cdot (b + c) \). Thus, \( 14 \cdot z + 14 \cdot 5 = 14 \cdot (z + 5) \).

Step 3: Verify the Expression

To ensure accuracy, distribute \(14\) back into the factored expression: \( 14 \cdot (z + 5) = 14 \cdot z + 14 \cdot 5 \). This confirms that the factorization is correct.

Key Concepts

Understanding Common Factors

A common factor is a number or variable that is shared by all terms in an algebraic expression. Identifying common factors is a key step in simplifying or factoring expressions because it allows you to combine terms or simplify processes.
Consider an expression like \( 14 \cdot z + 14 \cdot 5 \). Both of these terms are "multiples" of 14. Therefore, 14 is the common factor in this expression.
Why is recognizing a common factor useful?

• It makes complex expressions simpler.
• Helps in solving equations more efficiently.
• Is a crucial step in utilizing the distributive property.

Once a common factor is identified, you can use the distributive property to factor it out.

What is an Algebraic Expression?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols like \(+\), \(-\), \(\times\), and \(\div\). A crucial point about algebraic expressions is that they do not include an equal sign -- that would make them an equation.
In the expression \(14 \cdot z + 14 \cdot 5\), we see numbers (14 and 5), a variable \(z\), and the operation \(+\). This mixture of elements allows us to perform various algebraic operations such as simplification or factoring.
Why care about algebraic expressions?

• They form the basis of algebraic equations and inequalities.
• Understanding them is key to solving algebra problems.
• They help in modeling real-world situations mathematically.

By mastering algebraic expressions, you gain a tool to work through many types of mathematical problems.

The Process of Factoring

Factoring is the process of breaking down an expression into simpler "factor" pieces, much like counting by groups rather than individually. It involves identifying and "factoring out" a number or variable that multiplies each term.
Let's use our earlier example: \( 14 \cdot z + 14 \cdot 5 \). To factor this expression, the common factor (14) is identified and pulled out: \(14 \cdot (z + 5)\). This regrouping simplifies the expression, showing how the terms are connected.
Benefits of factoring include:

• Simplifying expressions for easier computation and readability.
• Solving algebraic equations by setting factors equal to zero.
• Understanding the structure and relationships within expressions.

Mastering factoring provides a pathway to deeper algebraic manipulation and problem-solving.