Problem 60 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(2+i)^{2}$$

Step-by-Step Solution

Answer

The product is \(3 + 4i\).

Step 1: Write the expression

The expression given is \((2 + i)^2\). We need to expand this expression by multiplying it by itself.

Step 2: Apply the formula for squaring a binomial

Recall that \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2\) and \(b = i\). We can apply this formula to find each term of the expansion.

Step 3: Calculate each term

First, calculate \(a^2 = 2^2 = 4\). Next, calculate \(2ab = 2 \times 2 \times i = 4i\). Finally, calculate \(b^2 = i^2 = -1\) since \(i^2 = -1\).

Step 4: Combine the terms

Add all the terms together: \(a^2 + 2ab + b^2 = 4 + 4i + (-1)\). Simplify this expression by combining the real numbers: \(4 - 1 = 3\). Thus, the expression simplifies to \(3 + 4i\).

Step 5: Write the final product in standard form

The standard form for a complex number is \(a + bi\). Therefore, the final product \(3 + 4i\) is already in standard form.

Key Concepts

Binomial Expansion

The binomial expansion is a systematic method used to expand expressions raised to a power, like \((a + b)^2\). This is important for simplifying the expression \((2 + i)^2\). Binomial expansion uses a specific formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

Here, \(a\) and \(b\) are elements of the binomial, which for our exercise are \(2\) and \(i\), respectively. The formula helps find each part of the expression when you distribute and multiply the terms. It's an efficient way to handle squared terms without manually multiplying, making calculations faster and easier.
By substituting our values into the formula, we break down the problem into smaller, more manageable parts. This results in \(a^2 = 4\), \(2ab = 4i\), and \(b^2 = -1\). Adding these helps get the final expanded form \(3 + 4i\).

Imaginary Unit

The imaginary unit, denoted as \(i\), is fundamental in complex numbers, where \(i\) represents the square root of \(-1\). It's crucial because no real number has a negative square, so \(i\) allows us to work within the realm of imaginary numbers.
In the exercise, the imaginary unit is part of our binomial term \((2+i)^2\), specifically the \(b\) in \((a+b)^2\). Knowing that \(i^2 = -1\) is pivotal for the expansion process.

• Calculating \(i^2\) in this problem results in \(-1\),

which is why \(b^2 = i^2 = -1\). This transformation allows us to better combine terms, such as turning \(4 + 4i + (-1)\) into \(3 + 4i\), and seamlessly integrate imaginary components into mathematical equations.

Standard Form of Complex Numbers

The standard form of complex numbers is represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Writing numbers in this form makes them uniform and easier to compare and manipulate.
When dealing with the result of a complex number operation, it is often required to express the final answer in standard form. For the problem \((2 + i)^2\), after expanding and simplifying, we ended up with \(3 + 4i\). Here, \(3\) is the real part, and \(4\) is the coefficient of the imaginary part.

• Any complex number can be easily interpreted in terms of its components,

thanks to this format. It is crucial for applications in engineering, physics, and other disciplines where complex numbers play a vital role. By ensuring the expression is in standard form, like our final product \(3 + 4i\), it adheres to mathematical convention and is neatly organized for further use.