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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1:36 minutes

Problem 15

Textbook Question

In Exercises 9–20, find each product and write the result in standard form.(3 + 5i)(3 - 5i)

Verified step by step guidance

Identify the expression to be multiplied: \((3 + 5i)(3 - 5i)\).

Recognize that this is a product of conjugates, which follows the pattern \((a + bi)(a - bi) = a^2 - (bi)^2\).

Apply the pattern: here, \(a = 3\) and \(b = 5\), so the expression becomes \(3^2 - (5i)^2\).

Calculate \(3^2\) which is \(9\).

Calculate \((5i)^2\) which is \(25i^2\), and remember that \(i^2 = -1\), so \(25i^2 = 25(-1) = -25\).

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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 and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations 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) to expand the product. For example, when multiplying (a + bi)(c + di), you multiply each part: ac, adi, bci, and bdi^2. Remember that i^2 equals -1, which helps simplify the result into standard form.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. When simplifying the product of complex numbers, the goal is to combine like terms and express the final result in this standard form, making it easier to interpret and use in further calculations.