Problem 63 - Elementary and Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i\) See Example 6 $$ (2+i)^{2} $$

Step-by-Step Solution

Answer

The product of \((2+i)^2\) is \(3+4i\).

Step 1: Identify the expression

We are tasked with multiplying \((2+i)^2\). This involves expanding the expression by multiplying two binomial expressions: \((2+i)(2+i)\).

Step 2: Expand the expression using FOIL

Use the FOIL method, which stands for First, Outer, Inner, Last, to expand. Calculate each part: - First: \((2)(2) = 4\) - Outer: \((2)(i) = 2i\) - Inner: \((i)(2) = 2i\) - Last: \((i)(i) = i^2\).

Step 3: Substitute and simplify

Combine the calculated terms: \(4 + 2i + 2i + i^2\). Remember that \(i^2 = -1\), so substitute \(-1\) for \(i^2\). The expression becomes: \(4 + 2i + 2i - 1\).

Step 4: Combine like terms

Add the real numbers and collect the imaginary parts: \(4 - 1 = 3\) and \(2i + 2i = 4i\). Thus, the expression simplifies to: \(3 + 4i\).

Step 5: Write the answer in \(a + bi\) form

The expression \(3 + 4i\) is already in the form of \(a + bi\). Thus the final result is that the multiplication \((2+i)^2\) equals \(3 + 4i\).

Key Concepts

Binomial Multiplication

Binomial multiplication is the process of multiplying two binomial expressions. A binomial is simply an expression that contains two terms. When you multiply two binomials, you have to ensure that each term in the first binomial gets multiplied by each term in the second binomial.
A practical example using complex numbers is multiplying \((2+i)(2+i)\). First, identify both parts of the binomials. Here, our two terms in each binomial are \(2\) and \(i\).
During multiplication, ensure every term in the first binomial is paired and multiplied with every term in the second binomial. This may initially look complicated with complex numbers, but by carefully applying rules and concepts, it becomes simpler.

• The multiplication of like terms \((2)(2)\) gives a real number value.
• The cross-products give the imaginary parts, such as \((2)(i)\) and \((i)(2)\), which will be combined later.
• Lastly, multiplying the imaginary parts \((i)(i)\) affects both the magnitude and the sign due to properties of the imaginary unit.

Ultimately, this method helps translate abstract algebraic work into concrete results, much like those achieved using the FOIL method.

FOIL Method

The FOIL Method is a helpful tool to simplify the process of multiplying binomial expressions. FOIL stands for First, Outer, Inner, and Last. It helps in visualizing and organizing the multiplication process.
Here's how the FOIL method works:

• **First**: Multiply the first terms of both binomials. For instance, multiplying \((2)(2)\) results in \(4\).
• **Outer**: Multiply the outer terms from each binomial. Using \((2)(i)\) gives you the term \(2i\).
• **Inner**: Multiply the inner terms. So here, \((i)(2)\) also results in \(2i\).
• **Last**: Multiply the last terms. The multiplication of \((i)(i)\) results in \(i^2\), which we know is equal to \(-1\).

When using the FOIL method, be mindful of simplifying the expression by combining like terms. In our example, the combination of the middle terms \(2i + 2i\) results in \(4i\). Additionally, substitute \(-1\) for \(i^2\) to refine the formula further.
The FOIL method remains a reliable approach, ensuring a structured experience in both arithmetic and algebra involving binomials.

Imaginary Unit

The imaginary unit, commonly represented as \(i\), is a fundamental part of complex numbers. It is defined as the square root of \(-1\). Complex numbers are expressed as \(a + bi\), where \(a\) is the real part and \(bi\) represents the imaginary part.
Key characteristics of the imaginary unit are:

• \(i^2\) is \\(-1\). This property is critical when dealing with squares of imaginary numbers.
• When multiplying imaginary terms, as in \((i)(i)\), it is transformed by substituting \(i^2\) with \(-1\), impacting both the sign and the nature of the product.

The presence of the imaginary unit in mathematical equations broadens the concept of numbers beyond the real number line, offering solutions to equations that would otherwise be unresolved.
Imaginary units, thus, facilitate the expression and understanding of complex numbers, playing a critical role in a wide array of real-world applications, from engineering to physics, and in mathematical problem-solving beyond basic arithmetic.