Problem 56 - Precalculus Student Solutions Manual 5th

Question

Find each product. Write the answer in standard form. $$(2+i)^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Write the expression to be expanded

The given expression is \( (2 + i)^2 \). This can be expanded using the formula for the square of a binomial: \( (a + b)^2 = a^2 + 2ab + b^2 \).

Step 2: Identify the terms

In this expression: \( a = 2 \) and \( b = i \).

Step 3: Expand the expression

Using the formula, \[ (2 + i)^2 = (2)^2 + 2(2)(i) + (i)^2 \].

Step 4: Calculate each term

Calculate each term in the expanded expression: \[ (2)^2 = 4 \] \[ 2(2)(i) = 4i \] \[ (i)^2 = i^2 = -1 \].

Step 5: Sum the terms

Combine all the calculated terms: \[ 4 + 4i - 1 \].

Step 6: Simplify

Combine like terms: \[ 4 - 1 + 4i = 3 + 4i \].

Key Concepts

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to the complex plane. Here, numbers have two parts: a real part and an imaginary part. For example, in the number \(2 + i\), 2 is the real part, and \(i\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

• **Real Part**: The real part of a complex number is a real number. In \(2 + i\), the real part is 2.
• **Imaginary Part**: This is the value multiplied by \(i\). In \(2 + i\), the imaginary part is 1.
• **Representation**: A complex number is generally written as \(a + bi\), where \(a\) and \(b\) are real numbers.

Complex numbers make it easier to solve equations that have no real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions because no real number squared gives -1, but it has two complex solutions: \(i\) and \(-i\).

Binomial Expansion

Binomial expansion involves expanding expressions that are raised to a power, represented as \((a + b)^n\). In our exercise, we expand \((2 + i)^2\).

• **Square of a Binomial**: When expanding \((a + b)^2\), we can use the formula \(a^2 + 2ab + b^2\).
• **Identify Variables**: For \((2 + i)^2\), let \(a = 2\) and \(b = i\).
• **Expand**: Plugging these values into the formula gives us \((2)^2 + 2(2)(i) + (i)^2\).
• **Evaluate Each Term**: Calculate each part separately: \((2)^2 = 4\), \(2(2)(i) = 4i\), and \((i)^2 = -1\).

This stepwise process makes it easy to handle more complex expressions involving binomials.

Standard Form

Writing complex numbers in standard form involves combining the real and imaginary parts. The standard form for complex numbers is \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

• **Combine Like Terms**: Start with the separately calculated real and imaginary parts. For example, from our previous calculations: \(4 + 4i - 1\).
• **Simplify**: Combine the real numbers: \(4 - 1 + 4i = 3 + 4i\).

This gives us the standard form. It's useful for easily identifying the real and imaginary parts of the complex number. Standard form is also crucial in performing operations like addition, subtraction, multiplication, and division of complex numbers.