Problem 108 - Beginning and Intermediate Algebra

Question

Find the product. $$2(k+5)^{2}$$

Step-by-Step Solution

Answer

The product of the given expression \(2(k+5)^{2}\) is \(2k^2 + 20k + 50\).

Step 1: Expand the squared binomial

First, let's expand the binomial (k+5)^2. Recall that squaring a binomial means multiplying the binomial by itself: \((k+5)^2 = (k+5)(k+5)\) Now, use the distributive property (FOIL method) to multiply the terms: \((k+5)(k+5) = k^2 + 5k + 5k + 25\) Combine the like terms: \(k^2 + 10k +25\)

Step 2: Multiply the expanded binomial by the constant 2

Now that we have expanded the binomial, let's multiply the result by the constant 2: \(2(k^2 + 10k + 25)\) Use the distributive property again to multiply each term by 2: \(2k^2 + 20k + 50\)

Step 3: Write the final expanded expression

Now that we have multiplied the expanded binomial by the constant, we can write the final expanded expression for the given problem: \(2(k+5)^2 = 2k^2 + 20k + 50\)

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a technique used to expand expressions raised to a power. When you see something like \((k+5)^2\), it's a squared binomial. Binomials are two-term expressions, and squaring means multiplying it by itself. The task is to transform this into a simpler form.
To start expanding, we need to write:

• Multiply the expression by itself: \((k+5)(k+5)\)

From here, the expansion process begins. This approach helps us break down complex terms into more manageable parts. Understanding how to handle binomial expressions can make solving problems involving powers more accessible. The next steps involve other important algebraic techniques like the distributive property, often used in tandem with methods like FOIL for further simplification.

Exploring the Distributive Property

The distributive property is a core algebraic principle. It allows you to multiply a single term by each term within a parenthesis. When working with our expression,\((k+5)(k+5)\), we use the distributive property to handle every term. This process ensures that each component of the binomial is accounted for in the multiplication.
The application involves:

• Multiplying each term in the first binomial by each term in the second binomial.

This gives us:

• \(k \cdot k + k \cdot 5 + 5 \cdot k + 5 \cdot 5\)
• Resulting in: \(k^2 + 5k + 5k + 25\)

After these calculations, the expression is combined through addition. The result is a simplified polynomial. The distributive property is essential not only in algebra but in higher-level mathematics as well, ensuring calculations can be performed systematically.

Utilizing the FOIL Method

The FOIL method is a specific case of the distributive property useful for multiplying two binomials. FOIL stands for First, Outer, Inner, Last—these are the terms you need to multiply when dealing with binomials like \((k+5)^2 = (k+5)(k+5)\).
This method breaks down as follows:

• **First:** Multiply the first terms: \(k \cdot k = k^2\)
• **Outer:** Multiply the outer terms: \(k \cdot 5 = 5k\)
• **Inner:** Multiply the inner terms: \(5 \cdot k = 5k\)
• **Last:** Multiply the last terms: \(5 \cdot 5 = 25\)

Together, these results provide:

• \(k^2 + 5k + 5k + 25\)
• Which simplifies to: \(k^2 + 10k + 25\)

The FOIL method is particularly useful when first learning about binomial multiplication. It offers an organized approach, ensuring no term is overlooked in the process.