Problem 6 - Algebra and Trigonometry

Question

1–8 ? Factor out the common factor. $$ (z+2)^{2}-5(z+2) $$

Step-by-Step Solution

Answer

The expression factors into \\( (z+2)(z-3) \\\).

Step 1: Identify the Common Factor

First, observe that both terms in the expression, \( (z+2)^2 \) and \( -5(z+2) \), have a common factor of \( (z+2) \). Thus, \( (z+2) \) is the common factor we want to factor out.

Step 2: Factor Out the Common Factor

Factor the expression by taking \( (z+2) \) out as a common factor. This means rewriting the expression as \[ (z+2)^{2}-5(z+2) = (z+2)((z+2)-5) \]

Step 3: Simplify the Expression Inside the Parenthesis

Now, simplify the expression within the parenthesis: \( (z+2) - 5 \). This results in \[ z + 2 - 5 = z - 3 \]

Step 4: Write the Factored Form

Substitute back the simplified expression within the brackets. The final expression is \( (z+2)(z-3) \), which is the expression factorized completely.

Key Concepts

Common Factor

In algebra, a common factor is a term that can be seen in multiple parts of an expression, as was seen with the expression \((z+2)^2 - 5(z+2)\). Identifying the common factor is the first crucial step in simplifying complex expressions.

Here, we noticed that \((z+2)\) appeared in both terms of the expression. This allows us to "factor out" \((z+2)\) from each part. It's like finding a common theme in a storyline and extracting it. By doing this, we drastically simplify the expression, making it much easier to manage. When factoring, always look for these shared terms across different parts.

• Locate common features in expressions.
• Use these shared features to simplify through factoring.

Factoring Techniques

Factoring is a key technique in algebra to simplify expressions by breaking them down into components. In the exercise, after identifying \((z+2)\) as a common factor, we factored \((z+2)\) out of both terms. When you do this, you're essentially re-writing the expression to expose underlying structures.

To illustrate, the initial expression \((z+2)^2 - 5(z+2)\) can be rewritten as \((z+2)((z+2)-5)\). This technique reduces the complexity by transforming the expression into a product of simpler terms. Always work to spot these opportunities to factor, as it can turn an intimidating equation into something more approachable.

• Rewrite expressions to highlight factoring opportunities.
• Factor out common terms to aid simplification.

Algebraic Simplification

Once the expression has been factored, we can take it a step further by simplifying what is inside the parentheses. Simplifying an algebraic expression involves combining like terms or performing arithmetic operations.

In our exercise, after factoring out \((z+2)\), we were left with \((z+2) - 5\) inside the parentheses. By simplifying this part of the equation, we achieve the expression \(z - 3\). The final factored form becomes \((z+2)(z-3)\), which is far simpler and cleaner.
Algebraic simplification makes complex problems manageable, aiding in deeper comprehension and solving equations accurately. Remember to always simplify as much as possible once you've factored an expression. This not only helps solve equations but also makes graphs and interpretations clearer.

• Combine like terms for further simplification.
• Aim to reach the simplest form of expression.