Problem 58 - Intermediate Algebra

Question

Multiply. Write all answers in the form a \(+b i.\) $$ (4-i)(2+i) $$

Step-by-Step Solution

Answer

The product is \(9 + 2i\).

Step 1: Distribute the Terms

The expression to solve is \((4-i)(2+i)\). We will distribute each term in the first complex number \((4-i)\) across each term in the second \((2+i)\). Based on the distributive property this gives us \[4\cdot2 + 4\cdot i - i\cdot2 - i\cdot i.\]

Step 2: Simplify the Products

Calculate each product from the expression:- \(4 \cdot 2 = 8\)- \(4 \cdot i = 4i\)- \(-i \cdot 2 = -2i\)- \(-i \cdot i = -i^2\)Remember that \(i^2 = -1\), so \(-i^2\) becomes \(-(-1) = 1\).

Step 3: Combine Like Terms

After performing the multiplications, we have the expression:\(8 + 4i - 2i + 1\). Now we combine like terms:- The real parts: \(8 + 1 = 9\).- The imaginary parts: \(4i - 2i = 2i\).Therefore, the expression simplifies to \(9 + 2i\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that involves multiplying a sum by multiplying each addend separately and then adding the products. When working with complex numbers, this principle becomes particularly useful. Consider the expression \( (4-i)(2+i) \). In applying the distributive property, each term of the first complex number \((4 - i)\) is multiplied separately by each term of the second complex number \((2 + i)\).
Let's break it down:

• Multiply \(4\) by both \(2\) and \(i\).
• Multiply \(-i\) by both \(2\) and \(i\).

This results in:

• \(4 \cdot 2 = 8\)
• \(4 \cdot i = 4i\)
• \(-i \cdot 2 = -2i\)
• \(-i \cdot i = -i^2\)

Understanding this step is crucial for simplifying algebraic expressions involving complex numbers, as it sets the stage for further simplification.

Imaginary Unit (i)

The imaginary unit \(i\) is a unique and intriguing concept in mathematics. It is defined as the square root of minus one: \(i^2 = -1\). This property helps in simplifying expressions involving complex numbers.
In our example, when simplifying the expression \(-i \cdot i = -i^2\), the presence of \(i^2\) requires us to replace \(i^2\) with \(-1\), as per its definition. Thus, \(-i^2\) becomes \(-(-1) = 1\).
The imaginary unit allows the extension of real numbers to complex numbers, which are essential in many fields like engineering and physics. Recognizing how to use \(i\) effectively is crucial when dealing with expressions that include imaginary numbers.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involving complex numbers is about combining like terms and reducing the expression to its simplest form. Starting with our expanded and distributed expression: \(8 + 4i - 2i + 1\), we identify and combine like terms.
First, add the real components:

• \(8 + 1 = 9\)

Next, combine the imaginary components:

• \(4i - 2i = 2i\)

By combining these results, the expression simplifies to \(9 + 2i\).
Through distillation of these terms, the process of simplification provides a neat and mathematically sound result. This step-by-step approach ensures clarity and accuracy, especially when handling more complex problems involving both real and imaginary parts.