Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 9

Chapter 5, Problem 9

In Exercises 9–20, find each product and write the result in standard form. −3i(7i − 5)

Verified step by step guidance

Recall that the imaginary unit \(i\) has the property \(i^2 = -1\).

Distribute \(-3i\) across the terms inside the parentheses: \(-3i \times 7i\) and \(-3i \times (-5)\).

Calculate each product separately: \(-3i \times 7i = -21i^2\) and \(-3i \times (-5) = 15i\).

Substitute \(i^2\) with \(-1\) in the expression \(-21i^2\) to get \(-21 \times (-1)\).

Simplify the expression to combine the real and imaginary parts, resulting in a number in the form \(a + bi\).

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Key Concepts

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

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1, with the property that i² = -1. It is fundamental in complex numbers and allows for the extension of the real number system to include solutions to equations like x² + 1 = 0.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and applying the rule i² = -1. Each term is multiplied carefully, combining like terms and simplifying to express the result in standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is written as a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. Expressing results in this form clearly separates real and imaginary components for easier interpretation.

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