all solutions to x^3=1 | Summary and Q&A

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February 4, 2018
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blackpenredpen
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all solutions to x^3=1

TL;DR

Learn how to find all the cube roots of the number one, including real and complex answers, using algebra and trigonometry methods.

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Questions & Answers

Q: How can you find the cube roots of unity?

To find the cube roots of unity, you can either solve the equation x^3 - 1 = 0 using algebra or use trigonometry and complex plane rotations.

Q: What is the algebraic method for finding the cube roots of unity?

The algebraic method involves factoring the equation x^3 - 1 = 0 using the difference of cubes formula and solving for x.

Q: How does the trigonometric method work for finding cube roots of unity?

In the trigonometric method, complex numbers are represented in the polar form and rotations on the complex plane are used to find the cube roots.

Q: Can you provide an example of finding the cube roots using the trigonometric method?

Yes, one of the cube roots of unity is found by calculating the cosine and sine of certain angles (such as 120 degrees and 240 degrees) and combining them with appropriate coefficients.

Summary & Key Takeaways

  • The video explains two methods to find cube roots of unity: algebra and trigonometry.

  • In the algebraic method, the equation x^3 - 1 = 0 is solved using the difference of cubes factorization.

  • In the trigonometric method, complex numbers are represented in the polar form and rotations on the complex plane are used to find the cube roots.

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