Problem 72 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (8-4 i)(7-2 i) $$

Step-by-Step Solution

Answer

The product is \(48 - 44i\).

Step 1: Apply the distributive property

To find the product of two complex numbers, we start by using the distributive property (also known as FOIL for binomials). Multiply each term in the first complex number by each term in the second complex number.\[(8 - 4i)(7 - 2i) = 8 \times 7 + 8 \times (-2i) + (-4i) \times 7 + (-4i) \times (-2i)\]

Step 2: Perform individual multiplications

Calculate the results of each multiplication:- \(8 \times 7 = 56\)- \(8 \times (-2i) = -16i\)- \(-4i \times 7 = -28i\)- \(-4i \times (-2i) = 8i^2\)

Step 3: Simplify using \(i^2 = -1\)

Substitute \(i^2 = -1\) to simplify the last term:\[8i^2 = 8(-1) = -8\]

Step 4: Combine like terms

Combine the real and imaginary parts separately:\[56 - 8 + (-16i - 28i) = (56 - 8) + (-16i - 28i) = 48 - 44i\]

Step 5: Express in standard form

The standard form of a complex number is \(a + bi\). The solution is already in this form:\[48 - 44i\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle where each term in one set is multiplied by every term in another set. This property allows for the expansion and simplification of expressions, making it essential when working with complex numbers. In the context of complex numbers, which have the form \(a + bi\), this principle is sometimes referred to as the FOIL method when dealing with binomials.

• It ensures each term in the first complex number is distributed or multiplied by each term in the second complex number. This is crucial in obtaining all components of a complex multiplication problem.
• The result is a combination of both real numbers and imaginary numbers. These components need to be simplified and combined in later steps.

Understanding the distributive property is the first key step in simplifying the product of complex numbers.

FOIL Method

FOIL is an acronym that stands for First, Outside, Inside, Last, and it is a specific application of the distributive property used for multiplying two binomials.

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the multiplication.
• Inside: Multiply the inner terms in the multiplication.
• Last: Multiply the last terms in each binomial.

This method helps in systematically organizing the multiplication process, making it easier to keep track of all the terms involved.
In the exercise given, - The "First" multiplication is \(8 \times 7\).- The "Outside" multiplication is \(8 \times (-2i)\).- The "Inside" multiplication is \((-4i) \times 7\).- The "Last" multiplication is \((-4i) \times (-2i)\). Each of these calculations is then combined, and like terms are simplified, leading to the result expressed in standard form.

Standard Form of a Complex Number

The standard form of a complex number is written as \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part.

• Real Part: This is the non-imaginary component of the complex number.
• Imaginary Part: This is the part that includes the imaginary unit \(i\), where \(i^2 = -1\).

In the example, simplifying \(-16i - 28i\) gives an imaginary component of \(-44i\). When the final real value \(48\) and the imaginary value \(-44i\) are fit into the \(a + bi\) format, we get \(48 - 44i\). This form is universally recognized and allows for consistent communication and computation when it comes to complex numbers.