Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Complex Number Division**
**Problem 4:**
\[ \frac{5 + 4i}{2 + i} \]
In this problem, we are tasked with dividing the complex number \(5 + 4i\) by \(2 + i\). To perform this division, we multiply the numerator and the denominator by the conjugate of the denominator.
**Steps:**
1. Identify the conjugate of the denominator. The conjugate of \(2 + i\) is \(2 - i\).
2. Multiply both the numerator and the denominator by the conjugate:
\[
\frac{(5 + 4i)(2 - i)}{(2 + i)(2 - i)}
\]
3. Simplify the expression. The denominator becomes a real number since:
\[
(2 + i)(2 - i) = 2^2 - i^2 = 4 + 1 = 5
\]
4. Multiply out the terms in the numerator:
\[
(5 + 4i)(2 - i) = 5 \cdot 2 - 5 \cdot i + 4i \cdot 2 - 4i \cdot i = 10 - 5i + 8i - 4i^2
\]
5. Simplify the expression:
Since \(i^2 = -1\), replace \(-4i^2\) with \(4\). The expression now is:
\[
10 - 5i + 8i + 4 = 14 + 3i
\]
6. Divide each term by the real number in the denominator:
\[
\frac{14 + 3i}{5} = \frac{14}{5} + \frac{3i}{5}
\]
Thus, the solution to the problem is:
\[
\frac{14}{5} + \frac{3}{5}i
\]
This solution explains how to divide complex numbers using the method of multiplying by the conjugate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9ca4a95-d733-4645-9dfa-b9a6044290e5%2Fdca3abd3-8a8c-4d9d-a45b-956583449976%2Fqyoi0yw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Complex Number Division**
**Problem 4:**
\[ \frac{5 + 4i}{2 + i} \]
In this problem, we are tasked with dividing the complex number \(5 + 4i\) by \(2 + i\). To perform this division, we multiply the numerator and the denominator by the conjugate of the denominator.
**Steps:**
1. Identify the conjugate of the denominator. The conjugate of \(2 + i\) is \(2 - i\).
2. Multiply both the numerator and the denominator by the conjugate:
\[
\frac{(5 + 4i)(2 - i)}{(2 + i)(2 - i)}
\]
3. Simplify the expression. The denominator becomes a real number since:
\[
(2 + i)(2 - i) = 2^2 - i^2 = 4 + 1 = 5
\]
4. Multiply out the terms in the numerator:
\[
(5 + 4i)(2 - i) = 5 \cdot 2 - 5 \cdot i + 4i \cdot 2 - 4i \cdot i = 10 - 5i + 8i - 4i^2
\]
5. Simplify the expression:
Since \(i^2 = -1\), replace \(-4i^2\) with \(4\). The expression now is:
\[
10 - 5i + 8i + 4 = 14 + 3i
\]
6. Divide each term by the real number in the denominator:
\[
\frac{14 + 3i}{5} = \frac{14}{5} + \frac{3i}{5}
\]
Thus, the solution to the problem is:
\[
\frac{14}{5} + \frac{3}{5}i
\]
This solution explains how to divide complex numbers using the method of multiplying by the conjugate.
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