Problem 69 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 5(x-2)+10 y(x-2) $$

Step-by-Step Solution

Answer

The expression factors to \(5(x-2)(1 + 2y)\).

Step 1: Identify the Common Factor

First, observe the given expression: \(5(x-2)+10y(x-2)\). Notice that \((x-2)\) is a common factor that appears in both terms of this expression.

Step 2: Factor Out the Common Factor

Since \((x-2)\) is present in both terms, factor \((x-2)\) out of the expression. This gives: \((x-2)(5 + 10y)\).

Step 3: Simplify the Expression Inside the Parentheses

Look at the expression \(5 + 10y\) inside the parentheses. Observe that there is a common factor of 5 in this expression. Factor the 5 out to get \(5(1 + 2y)\).

Step 4: Write the Fully Factored Form

Substitute \(5(1 + 2y)\) back into the expression: \((x-2)(5(1 + 2y))\). Simplify the expression: \(5(x-2)(1 + 2y)\). This is the fully factored form of the original expression.

Key Concepts

Common Factor

In algebra, identifying a common factor is a crucial step in simplifying expressions. A common factor is a term or number that appears in each part of an expression. By finding common factors, we can make complex expressions easier to work with.

To identify a common factor in an algebraic expression:

• Look at each term of the expression to find what they have in common.
• In our exercise, the term \((x-2)\) was present in both terms of the expression \(5(x-2) + 10y(x-2)\). Thus, \((x-2)\) is the common factor.

Finding common factors is essential in breaking down larger problems into simpler parts. It helps in reducing potential errors and streamlining further calculations.

Algebraic Simplification

Algebraic simplification is the process of making an expression easier to work with by reducing it to its simplest form. After identifying common factors, algebraic simplification usually involves combined terms, reducing fractions, or factoring expressions.

Let's see how it was applied in our problem:

• After factoring out \((x-2)\), the expression was \((x-2)(5 + 10y)\). This was simplified by noting that within the parentheses, \(5+10y\), there was another common factor: 5.
• By factoring out the 5, we simplified it to \(5(1 + 2y)\).

Simplifying expressions not only makes calculations easier but also helps to better understand the underlying structure of the expression.

Factored Form

The factored form of an expression is a product of its factors. It is the final output after identifying all common factors and simplifying each part of an expression. Representing an expression in factored form highlights its components, making it easier to solve equations or evaluate expressions.

In our example, the expression \(5(x-2)(1 + 2y)\) is the factored form. Here's why:

• We started by identifying \((x-2)\) as a common factor.
• Further simplified the expression \(5 + 10y\) inside the parentheses to \(5(1 + 2y)\).
• Combine these to express the whole as \(5(x-2)(1 + 2y)\).

This process demonstrates the power of combining factorization and simplification, helping us effortlessly reach the most straightforward representation of an expression.