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 5.1.59

Chapter 5, Problem 5.1.59

Evaluate x² − 2x + 2 for x = 1 + i.

Verified step by step guidance

Recognize that the problem involves evaluating the expression \(x^2 - 2x + 2\) where \(x\) is a complex number, specifically \(x = 1 + i\), with \(i\) being the imaginary unit satisfying \(i^2 = -1\).

Substitute \(x = 1 + i\) into the expression to get: \((1 + i)^2 - 2(1 + i) + 2\).

Expand the squared term using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \((1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i + i^2\).

Replace \(i^2\) with \(-1\) and simplify the expression inside the parentheses: \(1 + 2i + (-1) = 2i\).

Now, substitute back and simplify the entire expression step-by-step: \(2i - 2(1 + i) + 2\). Expand the multiplication and combine like terms to simplify the expression fully.

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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 of the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to perform arithmetic operations with complex numbers is essential for evaluating expressions involving them.

Substitution in Algebraic Expressions

Substitution involves replacing a variable in an expression with a given value. In this case, substituting the complex number x = 1 + i into the polynomial allows evaluation of the expression for that specific input.

Simplification of Expressions with Imaginary Units

Simplifying expressions with imaginary units requires applying the rule i² = -1 and combining like terms carefully. This process ensures the expression is reduced to its simplest form, often resulting in a complex number in standard form a + bi.

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Related Practice

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

x² + y² = 16

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

( −6 − √(−12)) / 48

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

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Textbook Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Line: Passes through (−2,4) and (1,7)

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