Problem 29 - Algebra 2

Question

Simplify. $$ (p+6)(p-4) $$

Step-by-Step Solution

Answer

The simplified expression is \( p^2 + 2p - 24 \).

Step 1: Distribute First Term

Start by distributing the first term of the first parentheses, which is \( p \), to each term in the second parentheses. This gives us \( p \times p + p \times (-4) \). Simplifying these expressions, we have \( p^2 - 4p \).

Step 2: Distribute Second Term

Next, distribute the second term of the first parentheses, which is \( 6 \), to each term in the second parentheses: \( 6 \times p + 6 \times (-4) \). Simplifying these calculations results in \( 6p - 24 \).

Step 3: Combine Like Terms

Now, combine all terms from the previous steps: \( p^2 - 4p + 6p - 24 \). Combine the like terms \(-4p + 6p\) to get \(2p\).

Step 4: Write the Final Expression

The simplified expression is \( p^2 + 2p - 24 \), after combining like terms and simplifying the expression.

Key Concepts

Distribution Method

The distribution method is a simple but powerful tool in algebra that helps in dealing with expressions involving parentheses. When you see an expression like \((p+6)(p-4)\), the distribution method assists in breaking down the multiplication into smaller, more manageable parts.

To apply this method, you should multiply each term inside the first set of parentheses by every term in the second set of parentheses.

• Start by distributing the first term: Multiply the first term inside the first parenthesis \(p\) by both terms inside the second parenthesis \((p-4)\). This gives us \(p \times p = p^2\) and \(p \times (-4) = -4p\).
• Next, distribute the second term: Do the same with the next term \(6\) in the first parenthesis, multiplying it by each term in the second parenthesis. This results in \(6 \times p = 6p\) and \(6 \times (-4) = -24\).

The distribution method allows us to transform complex multiplications into simpler parts that can then combine to form a new expression.

Simplifying Expressions

Simplifying expressions involves breaking down a math problem into its most basic components without changing its value. After distributing terms as shown previously, the expression might look complex, but simplification aims to make it easier to understand and solve.

When you simplify an expression like the one obtained after distribution \(p^2 - 4p + 6p - 24\), focus on rewriting it in the simplest terms possible.

• Keep constants and variables separate: Rearrange the terms logically, with terms of the same kind grouped together for ease of further operation.
• Handle operations step by step: Complete any arithmetic operations like addition and subtraction to simplify these internal components further.

These practices ensure that the expression is not only easier to evaluate but also easier to see the connections and potential solutions.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms in an algebraic expression that have the same variables and powers. This is a key step in simplifying expressions after using the distribution method.

Consider the result \(p^2 - 4p + 6p - 24\) from our distributed terms. The terms \(-4p\) and \(6p\) are like terms because they each have a single \(p\) raised to the power of one:

• Combine coefficients: Add or subtract the coefficients of like terms. Here, \(-4p + 6p = 2p\).
• Write the expression: Combine these with any other remaining terms like constants (in this case, \(-24\)). The final expression becomes \(p^2 + 2p - 24\).

By combining like terms, longer and more complicated expressions are simplified, making them easier to interpret and solve.