Problem 32 - Algebra A Combined Function

Question

Multiply. $$ (6 s+2)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(36s^2 + 24s + 4\).

Step 1: Recognize the format

The expression to be solved is \((6s + 2)^2\). This is a binomial raised to a power, which means it can be expanded using the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 6s\) and \(b = 2\).

Step 2: Square the first term

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), start with the first term \(a = 6s\). Calculate \((6s)^2 = 36s^2\). This is the first term in the expanded expression.

Step 3: Calculate twice the product of both terms

Now, find the middle term of the expansion by calculating \(2ab\). So we compute \(2 \times 6s \times 2 = 24s\). This is the second term in the expanded expression.

Step 4: Square the second term

Next, square the second term \(b = 2\). Calculate \(2^2 = 4\). This is the last term in the expanded expression.

Step 5: Write out the expanded expression

Compile the terms calculated in steps 2, 3, and 4 to form the complete expanded expression: \(36s^2 + 24s + 4\). Therefore, the expression \((6s + 2)^2\) is expanded as \(36s^2 + 24s + 4\).

Key Concepts

Binomial Expansion

Binomial expansions are a powerful technique in algebra used to simplify expressions where a binomial is raised to a power. Understanding how to expand these expressions forms the foundation for working with polynomials. A binomial is a two-term algebraic expression, such as \((a + b)\). When expanding \((a + b)^n\), especially for small values of \(n\), certain formulas like Pascal's triangle or the binomial theorem can be employed. However, the most straightforward method for small powers like 2, 3, or sometimes 4, is using patterns or known expansion formulas.
For example, when the power is 2, you might use the binomial square formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This process involves calculating each term separately and then adding them together. With practice, binomial expansion can become a quick process that adds a lot of efficiency to working with algebraic expressions.

Square of a Binomial

The square of a binomial refers to multiplying a binomial by itself, and it's a specific form of polynomial expansion. This is one of the core formulas every student of algebra learns because it frequently appears in mathematics.
The formula for the square of a binomial \((a + b)^2\) can be memorized as:

• First, square the first term: \(a^2\)
• Then, add twice the product of the two terms: \(2ab\)
• Finally, add the square of the second term: \(b^2\)

So for an expression like \((6s + 2)^2\), it follows these steps:

• Square the first term \((6s)^2\) which equals \(36s^2\)
• Calculate twice the product of \(6s\) and \(2\) which equals \(24s\)
• Square the second term \(2^2\) which equals \(4\)

Combining all these, the expanded form for \((6s + 2)^2\) is \(36s^2 + 24s + 4\). These steps illustrate how straightforward expanding a binomial square can be with practice.

Polynomial Multiplication

Polynomial multiplication extends beyond simple binomials and involves multiplying expressions with more than two terms. It includes efficiently handling larger sets of terms and organizing them into a simplified result.
Here are a few guidelines:

• Distribute each term from one polynomial across all terms of the other polynomial.
• Multiply each pair of terms and note their degree (for example, \(s^2\) when \(s \times s\)).
• Sum and combine like terms—those with the same degree—into one term.

While polynomial multiplication can get tricky with larger expressions, using a structured approach and keeping terms organized can make it easier. This method ensures you don't miss any terms or accidentally miscalculate. Developing these skills makes handling more complex algebraic problems much more manageable. By grasping the idea of multiplying each term sequentially, students can conquer even the intricate problems encountered in advanced mathematics.