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 17

Chapter 5, Problem 17

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

Verified step by step guidance

Recognize that the expression \((−5 + i)(−5 − i)\) is a product of two complex conjugates. Complex conjugates have the form \(a + bi\) and \(a - bi\).

Recall the formula for the product of complex conjugates: \((a + bi)(a - bi) = a^2 + b^2\). This results in a real number because the imaginary parts cancel out.

Identify \(a = -5\) and \(b = 1\) from the given expression \((−5 + i)(−5 − i)\).

Apply the formula by squaring \(a\) and \(b\) and then adding them: calculate \((-5)^2 + (1)^2\).

Write the result in standard form, which for complex numbers is \(x + yi\), but since the product of conjugates is real, the imaginary part will be zero.

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Key Concepts

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

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly as a sum of a real number and an imaginary number. Understanding this form is essential for interpreting and simplifying complex number expressions.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and applying the rule i² = -1. Each term in the first complex number is multiplied by each term in the second, and like terms are combined to simplify the expression.

Imaginary Unit Properties

The imaginary unit i is defined such that i² = -1. This property is crucial when simplifying products involving i, as it allows conversion of i² terms into real numbers, enabling the expression to be written in standard form.

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Related Practice

Textbook Question

In Exercises 9–20, find each product and write the result in standard form.

(3 + 5i)(3 − 5i)

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation.r = 2 + 2 cos θ

Textbook Question

In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)

Textbook Question

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (3, 4π/3)

Textbook Question

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−1, π)

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

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