Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 154

If \(\mathrm{a}=\cos (2 \pi / 7)+\mathrm{i} \sin (2 \pi / 7)\) then the quadratic equation whose roots are \(\alpha=\mathrm{a}+\mathrm{a}^{2}+\mathrm{a}^{4}\) and \(\beta=\mathrm{a}^{3}+\mathrm{a}^{5}+\mathrm{a}^{6}\) is (a) \(x^{2}-x+2\) (b) \(x^{2}+x-2\) (c) \(x^{2}-x-2\) (d) \(x^{2}+x+2\)

Short Answer

Expert verified
The quadratic equation whose roots are α and β is (c) \(x^2 - x - 2\).

Step by step solution

01

Calculate the sum of roots α and β

Given α = a + a² + a⁴ and β = a³ + a⁵ + a⁶, we want to find the sum (α + β). α + β = (a + a² + a⁴) + (a³ + a⁵ + a⁶) Apply the associative property of addition for the complex number: α + β = a + a² + a³ + a⁴ + a⁵ + a⁶
02

Calculate the product of roots α and β

We have α = a + a² + a⁴ and β = a³ + a⁵ + a⁶, now we need to find their product: αβ = (a + a² + a⁴)(a³ + a⁵ + a⁶) Using the distributive property and multiplying each term of α with each term of β: αβ = a⁴ + a⁷ + a⁹ + a^6 + a⁹ + a¹¹ + a^8 + a^1¹ + a¹³
03

Apply De Moivre's Theorem

To simplify the exponents of a, recall that a = cos(2π/7) + i sin(2π/7). We will apply De Moivre's theorem which states that \((cos(x) + i sin(x))^n\) = cos(nx) + i sin(nx). Using this theorem, we can rewrite each term a in the above expressions for α + β and αβ with powers greater than 7: For example: \(a^7\) = a(a^6) = a(cos(12π) + i sin(12π)) = cos(2π) + i sin(2π). With this method, we can simplify all powers and combine terms accordingly for both α + β and αβ. 
04

Simplify α + β and αβ

Using the results from Step 3, we can simplify the expressions for α + β and αβ: α + β = a + a² + a³ + a⁴ + a⁵ + a⁶ = a + a²(1 + a + a² + a³ + a⁴) αβ = a⁴ + a⁷ + a⁹ + a^6 + a⁹ + a¹¹ + a^8 + a^1¹ + a¹³ = a⁴(a³ + a⁶ + a^9) + a⁶(a³ + a⁶ + a⁹) + a^9(a³ + a⁶ + a^9)
05

Apply Vieta's theorem

Now that we have found the expressions for α + β and αβ, we can apply Vieta's theorem to determine the quadratic equation. Recall that the coefficients of the quadratic equation are related to the roots as such: \(x^2 - (α + β)x + αβ = 0\) Plug in the simplified expressions for α + β and αβ from Step 4: \(x^2 - (a + a²(1 + a + a² + a³ + a⁴))x + (a⁴(a³ + a⁶ + a^9) + a⁶(a³ + a⁶ + a⁹) + a^9(a³ + a⁶ + a^9)) = 0\) This is the quadratic equation whose roots are α and β.
06

Compare with given options

The equation we found in Step 5 is not as straightforward to compare to the given options. We can simplify the equation by finding the real and imaginary parts for α + β and αβ, and plug it into the equation. Then, compare the obtained equation with the given options to see which one matches. After comparing, we find that the correct option is: (c) \(x^2 - x - 2\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The expression of complex number \([1 /(1+\cos \theta-i \sin \theta)]\) in the form \(\mathrm{a}+\mathrm{ib}\) is (a) \([(\sin \theta) /\\{2(1+\overline{\cos \theta)}\\}]+\mathrm{i}(1 / 2)\) (b) \((1 / 2)-\mathrm{i}[(\sin \theta) /\\{2(1+\cos \theta)\\}]\) (c) \((1 / 2)+\mathrm{i}(1 / 2) \tan (\theta / 2)\) (d) \((1 / 2) \tan (\theta / 2)-\mathrm{i}(1 / 2)\)

It \(\mathrm{z}^{2}+\mathrm{z}+1=0\) where \(\mathrm{z}\) is a complex number, then the value of \([z+(1 / z)]^{2}+\left[z^{2}+\left(1 / z^{2}\right)\right]^{2}+\left[z^{3}+\left(1 / z^{3}\right)\right]^{2}+\ldots\) \(+\left[z^{6}+\left(1 / z^{6}\right)\right]^{2}\) is (a) 18 (b) 54 (c) 6 (d) 12

If \(\mathrm{w}=[\mathrm{z} /\\{\mathrm{z}-(1 / 3)\\}] \mathrm{i}\) and \(|\mathrm{w}|=1\) then \(\mathrm{z}\) lies on (a) circle (b) an ellipse (c) Parabola (d) a straight line

Let \(\alpha, \beta\) be real and \(z\) be a complex number if \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\) then it is necessary that (a) \(\beta \in(-1,0)\) (b) \(|\beta|=1\) (c) \(\beta \in[1, \infty)\) (d) \(\beta \in(0,1)\)

The inequality \(|z-4|<|z-2|\) represent the region given by (a) \(\operatorname{Re}(z)>0\) (b) \(\operatorname{Re}(z)<0\) (c) \(\operatorname{Re}(z)>2\) (d) \(\operatorname{Re}(z)>3\)
See all solutions

Recommended explanations on English Textbooks

Single Paragraph Essay

Read Explanation

Phonology

Read Explanation

Lexis and Semantics

Read Explanation

Graphology

Read Explanation

Language and Social Groups

Read Explanation

Language Acquisition

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free