Problem 122 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 2 x+2 y+2 z $$

Step-by-Step Solution

Answer

The expression simplifies to \(2(x + y + z)\).

Step 1: Identify Common Factors

Examine the expression \(2x + 2y + 2z\). Notice that each term has a common factor of 2. The terms are \(2x\), \(2y\), and \(2z\), all of which can be divided by 2.

Step 2: Factor Out the Common Factor

Factor the common factor of 2 out of the expression. The expression \(2x + 2y + 2z\) can be rewritten by factoring the 2 out as follows: \(2(x + y + z)\).

Step 3: Simplified Expression

The expression \(2(x + y + z)\) is now fully simplified; there are no further common factors or operations needed. The simplified form of the original expression is \(2(x + y + z)\).

Key Concepts

Common Factors

In algebraic expressions, identifying common factors is a key step towards simplification. The common factor is a number or variable that divides all terms in an expression evenly. This can often make seemingly complicated equations much simpler to handle. In our example, the expression provided is \(2x + 2y + 2z\). Here, the common factor among all the terms is 2. Each term \(2x\), \(2y\), and \(2z\) contains a factor of 2.
To identify a common factor:

• Look at each term within the expression.
• Identify if there is a number or variable that is present in each term.
• Ensure that this factor divides each term without leaving a remainder.

By factoring out the common factor, you essentially reverse the process of distribution, which we'll discuss next. This makes the expression more concise and focused, preparing it for further operations or interpretations.

Distribution

Distribution in algebra involves distributing a multiplied number across terms in parentheses. It's essentially the opposite of factoring. In expressions like our starting expression, the process is often done in reverse during simplification.The act of pulling a common factor, like 2 from \(2x + 2y + 2z\), is also a form of distribution. In this process:

• Recognize the common factor across all terms.
• Rewrite the equation by factoring out the common number, placing what's left inside parentheses.
• This shifts the emphasis from individual terms to a collective group inside the parentheses.

By factoring out 2, the expression becomes \(2(x + y + z)\). Distribution ensures that mathematical operations apply properly across expressions, allowing you to handle equations systematically.

Simplification of Algebraic Expressions

Simplification of algebraic expressions is a fundamental skill in algebra that makes complex expressions easier to work with and understand. The goal is to reduce the expression to its simplest form, where no further factoring or arithmetic changes can be applied.In the expression we've been simplifying, \(2x + 2y + 2z\), once the common factor of 2 is factored out and distributed, it reduces to \(2(x + y + z)\). This is the simplest form of the expression:

• All terms share the common factor, making it eligible for factoring out.
• Expression inside the parentheses, \((x + y + z)\), cannot be simplified further due to differing variables.

By simplifying expressions, calculations and predictions become easier in later algebraic operations. You can see relationships between variables more clearly, assisting in solving equations effectively.