Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:30 minutes

Problem 25

Textbook Question

Perform each operation. Write answers in standard form. (5-11i)(5+11i)

Verified step by step guidance

Recognize that the expression \((5-11i)(5+11i)\) is a product of two complex conjugates.

Recall the formula for multiplying complex conjugates: \((a - bi)(a + bi) = a^2 + b^2\).

Identify \(a = 5\) and \(b = 11\) from the expression \((5-11i)(5+11i)\).

Apply the formula: calculate \(a^2 = 5^2\) and \(b^2 = 11^2\).

Add the results from the previous step to get the expression in standard form: \(a^2 + b^2\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations involving them, such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. For example, when multiplying (a + bi)(c + di), you calculate ac, adi, bci, and bdi^2, remembering that i^2 = -1. This process results in a new complex number in the form of (ac - bd) + (ad + bc)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. When performing operations on complex numbers, it is important to simplify the result into this form for clarity and consistency. This involves combining real and imaginary parts and ensuring that any instances of i^2 are replaced with -1.