Problem 48 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The product is \(-10i\).

Step 1: Apply the Distributive Property

Multiply each term in the first binomial with each term in the second binomial. The expression is \((2-4i)(2-i)\).Hence, 1. Multiply \(2\) with \(2\) to get \(4\).2. Multiply \(2\) with \(-i\) to get \(-2i\).3. Multiply \(-4i\) with \(2\) to get \(-8i\).4. Multiply \(-4i\) with \(-i\) to get \(4i^2\).

Step 2: Combine Like Terms

Combine all the real and imaginary parts:1. The real parts are \(4 + 4i^2\).2. The imaginary parts are \(-2i - 8i\), simplifying to \(-10i\).

Step 3: Use the Property of \(i^2\)

Remember that \(i^2 = -1\). Substitute \(i^2\) with \(-1\) in the expression.So, \(4i^2 = 4 \times -1 = -4\).The new expression becomes \((4 - 4) - 10i\).

Step 4: Simplify the Expression

Simplify the expression:\(4 - 4 = 0\).Thus, the expression simplifies to \(0 - 10i\) or simply \(-10i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept not only in real numbers but also in complex numbers. When you apply the distributive property, you essentially take each term in one set of parentheses and multiply it by each term in the other parentheses. This step-by-step multiplication ensures that every possible product combination is considered.

In the context of complex numbers, consider the problem \((2-4i)(2-i)\). Here is how you can break it down:

• Multiply the first term of the first binomial (2) by both terms of the second binomial (2 and -i) resulting in \(4\) and \(-2i\).
• Do the same with the second term in the first binomial (-4i), yielding \(-8i\) and \(4i^2\).

Remember, every product should be accounted for, as missing any can result in an incorrect solution.
Practicing the distributive property with complex numbers will enhance your algebraic skills and make it easier to simplify complex expressions later.

Imaginary Unit

Complex numbers incorporate a special unit known as the imaginary unit, denoted by \(i\). This imaginary unit represents the square root of -1. So, by definition, \(i^2 = -1\). Imaginary numbers extend our numeric system to solve equations that have no solutions among real numbers.

In the problem \((2-4i)(2-i)\), we see the term \(4i^2\). To convert this into a form that we can easily work with, we apply the property \(i^2 = -1\):

• Multiply \(4i^2\) by substituting \(i^2\) with \(-1\), resulting in \(4 \times -1 = -4\).

This transformation is crucial because it allows us to handle the imaginary part of complex equations as real numbers and helps in simplifying the total expression.

Simplifying Expressions

Once you've applied the distributive property and understood the role of the imaginary unit, you can focus on simplifying expressions. Simplifying complex expressions involves combining like terms—both real and imaginary parts.

For instance, in our problem, after distributing and substituting \(i^2 = -1\), you obtain an expression like \(4 + 4i^2 -10i\). Simplifying involves these steps:

• First, substitute \(4i^2\) with \(-4\) since we know \(i^2 = -1\).
• This changes the expression to \(4 - 4 - 10i\), which simplifies to \((0 - 10i)\).

This simplification effectively reduces the expression to the imaginary unit form, reflecting how combining like terms can simplify and solve complex number operations.