Problem 90 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((2-3 i)(5+2 i)\)

Step-by-Step Solution

Answer

The expression simplifies to 16 - 11i.

Step 1: Apply the Distributive Property

To simplify (2-3i)(5+2i), we will first use the distributive property (also known as FOIL for binomials). Distribute each term in the first complex number across the terms in the second: 1. Multiply the first terms: 2 * 5 = 10 2. Multiply the outer terms: 2 * 2i = 4i 3. Multiply the inner terms: -3i * 5 = -15i 4. Multiply the last terms: -3i * 2i = -6i^2

Step 2: Combine Like Terms

Add the real and imaginary parts together separately. Combine the real numbers and the imaginary numbers: Real part: 10 + (-6i^2) Imaginary part: 4i - 15i

Step 3: Simplify Using the Property of i

Remember that i^2 = -1. Simplify -6i^2: -6(-1) = 6. Now substitute this back in: Real part: 10 + 6 = 16. Imaginary part: 4i - 15i = -11i.

Step 4: Write the Expression in Standard Form

Combine the results of the real and imaginary parts: Thus, (2-3i)(5+2i) = 16 - 11i. The expression is now in the standard form a+bi.

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra that makes it possible to multiply a single term by multiple terms within a parenthesis. In the context of complex numbers, this property helps us perform multiplication between two binomials. Let’s break it down with a simple expression:Given two complex numbers, for instance

• (a + bi) and (c + di),

the distributive property allows us to expand the product as follows:

• \( (a + bi)(c + di) = a(c + di) + bi(c + di) \)

This step-by-step application helps to simplify the expression systematically by distributing each component of the first complex number over every element of the second.
These distributed parts then sum up to give the product of the two complex numbers as another complex number. When we distribute each part of

• (2 - 3i)(5 + 2i),

we see this method in action, where each term in the first binomial is multiplied by each term in the second.

FOIL Method

The FOIL method is a particular instance of the distributive property that is especially useful in multiplying two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last. These terms guide us through multiplying each part systematically so nothing is left out.
Here's how the FOIL method works with the expression

• (2 - 3i)(5 + 2i):

• First: Multiply the first terms: 2 * 5 = 10
• Outer: Multiply the outer terms: 2 * 2i = 4i
• Inner: Multiply the inner terms: -3i * 5 = -15i
• Last: Multiply the last terms: -3i * 2i = -6i^2

By using the FOIL method, we can efficiently multiply binomials without missing any terms along the way. This method shines in its simplicity, making it easy to remember and apply, especially with complex numbers that involve the imaginary unit \(i\).

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined such that\(i^2 = -1\). This form succinctly separates the real and imaginary components, making the complex number easily understandable.
When we simplify

• \((2 - 3i)(5 + 2i)\)

through the steps of multiplication, we end up with a result like 16 - 11i.
In this case:

• \(a = 16\) represents the real part, and
• \(-11i\) is the imaginary component.

Once we know these two separate parts, we can clearly see the entire expression in standard form. It’s crucial for solving, graphing, and interpreting the behavior of complex numbers in various mathematical contexts.