Problem 60 - Elementary Algebra

Question

Factor out the GCF. $$ (x+6)-3 x 2(x+6) $$

Step-by-Step Solution

Answer

The factored form is \(-5(x+6)\).

Step 1: Identify Common Terms

Look at the terms in the expression \((x + 6) - 3 \times 2(x + 6)\). Notice that \((x+6)\) appears in both terms.

Step 2: Factor out the Common Factor

Realize that \((x+6)\) is the greatest common factor (GCF). Rewrite the expression by factoring out \((x+6)\):\[(x+6)(1 - 3 \times 2)\]

Step 3: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. Calculate \(3 \times 2\), which equals 6. Then, subtract 6 from 1:\[(1 - 6) = -5\]

Step 4: Write the Final Factored Expression

Substitute the simplified result back into the expression:\[(x+6)(-5)\]Thus, the factored form of the expression is \(-5(x+6)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor, often abbreviated as GCF, is a fundamental concept in factoring algebraic expressions. The GCF is the largest factor shared by all terms in an expression. In the context of the exercise, we look for the largest expression that can be evenly divided from each term. Here, the expression \((x+6)\) appears in both terms, making it the GCF.When finding the GCF:

• Identify common terms or factors in the given expression.
• Check each term to ensure that any chosen factor, like \((x+6)\), can be factored out evenly.
• If multiple common factors are present, the GCF is the product of these factors.

Finding the GCF simplifies the expression, allowing for easier manipulation and solution analysis.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operation symbols (such as \(+, -\), and multiplication). They represent mathematical relationships and can be simplified, factored, or expanded to solve equations or inequalities.In our exercise, the original expression was \((x+6) - 3 \times 2(x+6)\). Here are some components to recognize in algebraic expressions:

• **Terms**: These are the parts of the expression separated by a \( + \) or \( - \) sign. In our case, \((x+6)\) and \(-3 \times 2(x+6)\) are terms.
• **Coefficients**: Numbers multiplying a variable, for example, \(-3 \times 2\) is a coefficient of the term \((x+6)\).
• **Constants**: Fixed numbers, like the \(6\) in \((x+6)\), which do not change.

This understanding forms the foundation for performing operations like factoring and simplifying these expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with or understand. In our exercise, after factoring out the GCF \((x+6)\), we simplify what's inside the parentheses.Here's the process for simplifying expressions like ours:

• After isolating the common factor, perform any arithmetic operations inside the parentheses. We had \(1 - 3 \times 2\).
• Calculate these operations: \(3 \times 2 = 6\), then \(1 - 6 = -5\).
• Combine like terms to arrive at a thoroughly simplified expression, in this case, \((x+6)(-5)\).

Simplifying expressions helps clarify the structure and can reveal solutions or solutions' pathways, facilitating further mathematical operations or solutions.