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

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College Algebra

1. Equations & Inequalities

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1:50 minutes

Problem 55

Textbook Question

Find each product. Write answers in standard form. (2+i)(3-2i)

Verified step by step guidance

Identify the expression to be multiplied: \((2+i)(3-2i)\).

Apply the distributive property (also known as the FOIL method for binomials): Multiply each term in the first binomial by each term in the second binomial.

Calculate the products: \(2 \times 3\), \(2 \times -2i\), \(i \times 3\), and \(i \times -2i\).

Combine the real parts and the imaginary parts separately.

Remember that \(i^2 = -1\) and use this to simplify the expression to standard form \(a + bi\).

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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 combine the real and imaginary parts. For example, when multiplying (a + bi)(c + di), you calculate ac, ad, bc, and bd, and then combine like terms, remembering that i^2 = -1 to simplify the result.

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 with complex numbers, it is important to simplify the result into this form for clarity and consistency, ensuring that the real and imaginary parts are clearly identified.