Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Powers and Roots of Complex Numbers

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Find The Roots Of Complex Numbers

Solve for $z^{3}-z^{2}+z=0$ $\displaystyle z^{3}-z^{2}+z=0$ (there may be more than one solution).

Possible Answers:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

$0,\frac{1}{2},\pm \sqrt{3i}$ $\displaystyle 0,\frac{1}{2},\pm \sqrt{3i}$

$0,1,\frac{1}{2}$ $\displaystyle 0,1,\frac{1}{2}$

$\displaystyle 0$

$0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$

Correct answer:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula, which is $\frac{-b\pm \sqrt{b^{2}4ac}}{2a}$ $\displaystyle \frac{-b\pm \sqrt{b^{2}4ac}}{2a}$

$z^{3}-z^{2}+z\rightarrow z\left ( z^{2}-z+1 \right )$ $\displaystyle z^{3}-z^{2}+z\rightarrow z\left ( z^{2}-z+1 \right )$ and now setting both $\displaystyle z$ and $z^{2}-z+1$ $\displaystyle z^{2}-z+1$ equal to zero we end up with the answers z=0 $\displaystyle z=0$ and $z=\frac{1\pm 3i}{2}$ $\displaystyle z=\frac{1\pm 3i}{2}$ and so the correct answer is $0,\frac{1}{2},\pm \frac{2\sqrt{3i}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{2\sqrt{3i}}{2}$ .

Report an Error

Example Question #11 : Find The Roots Of Complex Numbers

Solve for all possible solutions to the quadratic expression: $m^{2}-6m+12=0$ $\displaystyle m^{2}-6m+12=0$

Possible Answers:

$3\pm \sqrt{3}$ $\displaystyle 3\pm \sqrt{3}$

$3\pm \sqrt{3i}$ $\displaystyle 3\pm \sqrt{3i}$

$3\pm 3$ $\displaystyle 3\pm 3$

$3\pm 3i$ $\displaystyle 3\pm 3i$

Correct answer:

$3\pm \sqrt{3i}$ $\displaystyle 3\pm \sqrt{3i}$

Explanation:

Solve for complex values of $\displaystyle m$ using the quadratic formula: $6\pm\sqrt{36-(4)(1)(12)} = \frac{6\pm\sqrt{-12}}{2}=\frac{6\pm2\sqrt{3}i}{2}=3\pm\sqrt{3}i$ $\displaystyle 6\pm\sqrt{36-(4)(1)(12)} = \frac{6\pm\sqrt{-12}}{2}=\frac{6\pm2\sqrt{3}i}{2}=3\pm\sqrt{3}i$ .

Report an Error

Example Question #11 : Find The Roots Of Complex Numbers

Solve for $z^{3}-z^{2}+z=0$ $\displaystyle z^{3}-z^{2}+z=0$ (there may be more than one solution).

Possible Answers:

$0,1,\frac{1}{2}$ $\displaystyle 0,1,\frac{1}{2}$

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

$0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$

$0,\frac{1}{2},\pm \sqrt{3i}$ $\displaystyle 0,\frac{1}{2},\pm \sqrt{3i}$

$\displaystyle 0$

Correct answer:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$ $\displaystyle 0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

Explanation:

To solve for the roots, just set equal to zero and solve for $\displaystyle z$ using the quadratic formula ( $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$ $\displaystyle \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$ ): $z^{3}-z^{2}+z\rightarrow z(z^{2}-z+1)$ $\displaystyle z^{3}-z^{2}+z\rightarrow z(z^{2}-z+1)$ and now setting both $\displaystyle z$ and $z^{2}-z+1$ $\displaystyle z^{2}-z+1$ equal to zero we end up with the answers z=0 $\displaystyle z=0$ and $z=\frac{1\pm 3i}{2}$ $\displaystyle z=\frac{1\pm 3i}{2}$ .

Report an Error

View Pre-Calculus Tutors

Usman
Certified Tutor

New York University, Bachelor in Arts, Political Science and Government.

View Pre-Calculus Tutors

Sara
Certified Tutor

University of Houston, Bachelor of Science, Psychology.

View Pre-Calculus Tutors

Jane
Certified Tutor

Johns Hopkins University, Doctor of Philosophy, Chemical and Biomolecular Engineering.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Philadelphia, Algebra Tutors in San Diego, ACT Tutors in Dallas Fort Worth, Spanish Tutors in San Diego, MCAT Tutors in Washington DC, Statistics Tutors in Miami, GRE Tutors in Boston, Spanish Tutors in New York City, MCAT Tutors in Houston, ACT Tutors in Seattle

Popular Courses & Classes

Spanish Courses & Classes in Atlanta, GRE Courses & Classes in Washington DC, GRE Courses & Classes in New York City, LSAT Courses & Classes in Denver, ACT Courses & Classes in Washington DC, ISEE Courses & Classes in New York City, MCAT Courses & Classes in Chicago, LSAT Courses & Classes in Los Angeles, SAT Courses & Classes in Seattle, ISEE Courses & Classes in Houston

Popular Test Prep

ACT Test Prep in Washington DC, MCAT Test Prep in Los Angeles, GRE Test Prep in New York City, MCAT Test Prep in Chicago, MCAT Test Prep in New York City, ACT Test Prep in Los Angeles, SAT Test Prep in Seattle, ACT Test Prep in San Francisco-Bay Area, GRE Test Prep in San Francisco-Bay Area, SSAT Test Prep in Atlanta

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Powers and Roots of Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #11 : Find The Roots Of Complex Numbers

Example Question #11 : Find The Roots Of Complex Numbers

Example Question #11 : Find The Roots Of Complex Numbers

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: