Problem 98 - Elementary Algebra

Question

Problem: Simplify: \((3 p)(p+4)(p+2)\) $$ \text { Incorrect Answer: } \begin{aligned} &(3 p)(p+4)(p+2) \\ &=3 p(p+4)+3 p(p+2) \\ &=3 p(p)+3 p(4)+3 p(p)+3 p(2) \\ &=3 p^{2}+12 p+3 p^{2}+6 p \\ &=6 p^{2}+18 p \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \(3p^3 + 18p^2 + 24p\).

Step 1: Identify the Expression

The given expression is \( (3p)(p+4)(p+2) \). The task is to simplify this expression correctly.

Step 2: Apply the Distributive Property

Multiply \( 3p \) by each term in the first binomial \( (p+4) \): \[ 3p(p+4) = 3p \times p + 3p \times 4 \] Simplifying gives: \[ 3p^2 + 12p \]

Step 3: Distribute the Result to the Second Binomial

Next, multiply \(3p^2 + 12p \) by each term in the second binomial \( (p+2) \): \[ (3p^2 + 12p)(p+2) \]

Step 4: Expand the Expression

Use the distributive property again to expand: \[ 3p^2 \times p + 3p^2 \times 2 + 12p \times p + 12p \times 2 \] Which simplifies to: \[ 3p^3 + 6p^2 + 12p^2 + 24p \]

Step 5: Combine Like Terms

Combine the like terms to simplify the expression: \[ 3p^3 + 18p^2 + 24p \]

Key Concepts

Distributive Property

The distributive property is crucial in simplifying algebraic expressions. It allows you to multiply a single term by each term within parentheses. For example, given an expression like \((3p)(p + 4)\), we can distribute \(3p \) to \(p\) and \(4\) individually.\[3p(p+4) = 3p \times p + 3p \times 4\]The result would then be:\[3p^2 + 12p\]This method helps break down complex expressions into simpler, manageable parts. Always remember that the distributive property involves multiplication over addition or subtraction within parentheses.

Combining Like Terms

Combining like terms involves simplifying expressions by merging terms that have the same variables raised to the same power. For instance, after distributing in the previous example, the resulting expression can be further simplified by combining like terms.\[3p^3 + 6p^2 + 12p^2 + 24p\]The terms \(6p^2\) and \(12p^2\) are like terms and can be combined:\[6p^2 + 12p^2 = 18p^2\]Thus, the simplified expression becomes: \[3p^3 + 18p^2 + 24p\]Combining like terms reduces the expression to its simplest form, making it easier to understand and work with.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another. Let's consider our expression \(3p(p + 4)(p + 2)\). First, we use the distributive property to handle \(3p(p + 4)\):\[3p(p + 4) = 3p^2 + 12p\]Next, we multiply the result by the second binomial \(p + 2\):\[(3p^2 + 12p)(p + 2)\]Using distributive property again: \[\begin{aligned}3p^2 \times p & = 3p^3\ 3p^2 \times 2 & = 6p^2\ 12p \times p & = 12p^2\ 12p \times 2 & = 24p\end{aligned}\]Now, we have: \[3p^3 + 6p^2 + 12p^2 + 24p\]Combining like terms, the final simplified version of the expression becomes: \[3p^3 + 18p^2 + 24p\]Through the polynomial multiplication process, we've correctly simplified the expression step-by-step. This process ensures that every term is accounted for and simplifies your algebraic work.