Problem 69 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(3+2 i)(5+4 i)$$

Step-by-Step Solution

Answer

The product is \(7 + 22i\).

Step 1: Apply the distributive property

To find the product of two complex numbers, apply the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first complex number by each term in the second complex number:\[(3 + 2i)(5 + 4i)\]is expanded as:\[3 \times 5 + 3 \times 4i + 2i \times 5 + 2i \times 4i\]

Step 2: Perform each multiplication

Calculate each multiplication from the expansion:1. \(3 \times 5 = 15\)2. \(3 \times 4i = 12i\)3. \(2i \times 5 = 10i\)4. \(2i \times 4i = 8i^2\)

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

Since \(i^2 = -1\), replace \(8i^2\) with \(-8\). Thus, the expression becomes:\[15 + 12i + 10i - 8\]

Step 4: Combine like terms

Combine the real parts and the imaginary parts separately:1. Real parts: \(15 - 8 = 7\)2. Imaginary parts: \(12i + 10i = 22i\)This results in the complex number \(7 + 22i\).

Step 5: Express in standard form

The final answer, in standard form (which is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part), is:\[7 + 22i\]

Key Concepts

Distributive Property

The distributive property is a key concept for multiplying complex numbers, just as it is for real numbers. In the context of complex numbers, it allows us to multiply two complex numbers by distributing each part of one number to each part of the other. This is how we fully expand the product.

For example, to find the product \[(3 + 2i)(5 + 4i),\]we use the distributive property by multiplying each term in the first complex number \((3 + 2i)\) by each term in the second complex number \((5 + 4i)\).

This means:

• Multiply \(3\) with \(5\) and \(4i\).
• Multiply \(2i\) with \(5\) and \(4i\).

This detailed breakdown results in the four products used to combine into one final expression.

FOIL Method

The FOIL method is a specialized version of the distributive property that is particularly useful for multiplying complex numbers expressed as binomial expressions. FOIL stands for First, Outer, Inner, Last, referring to the terms you need to multiply to simplify a product of two binomials.

Applying FOIL to the expression \((3 + 2i)(5 + 4i)\) involves the following steps:

• First: Multiply the first terms: \(3 \times 5 = 15\).
• Outer: Multiply the outer terms: \(3 \times 4i = 12i\).
• Inner: Multiply the inner terms: \(2i \times 5 = 10i\).
• Last: Multiply the last terms: \(2i \times 4i = 8i^2\).

Each product reflects a crucial step towards simplifying the expression into a standard complex number form.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the foundation of complex numbers. It is defined by the property that \(i^2 = -1\). This characteristic is critical when you work with the products of complex numbers, allowing us to replace products of \(i^2\) with \(-1\) during calculations.

In the multiplication process for \((3 + 2i)(5 + 4i)\), at some point you'll encounter an expression like \(8i^2\). Using the property \(i^2 = -1\), you transform \(8i^2\) into \(-8\).

Recognizing and correctly using the imaginary unit helps you move from an intermediate complex form to a simpler expression ready for combination into the standard form.

Standard Form of a Complex Number

The standard form of a complex number is crucial for communicating and manipulating complex expressions efficiently. It follows the format \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part.

After distributing, applying FOIL, and simplifying with \(i^2\), the expression from the multiplication of \((3 + 2i)(5 + 4i)\) transforms into \(7 + 22i\).

This final value expresses the complex number in its standard form, making it easy to see:

• The real component: \(7\)
• The imaginary component: \(22i\)

Being comfortable with rewriting complex numbers in this standard form is essential for understanding their properties and for further calculations.