# Need help with few questions

Algebra

### Need help with few questions

Hi I need help in solving the following equation. This question is Picked up from subject BCS12 Page 6. I have solved it on paper but I am not able to understand some logic. Please see the paper for further details.

The answer given in Book is :

$$W^{2}$$+W=-1

because

$$W^{2}$$+W+1=0

For my question please refer to the attachment. I have tried solving but not clear with the logic
Attachments
Unit 1 Question.jpg (67.16 KiB) Viewed 247 times
sunny

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### Re: Need help with few questions

Unable to post 2 images in same post so posting img from book which contains answer
Attachments
book question and answer unit 1.png (6.26 KiB) Viewed 245 times

sunny

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### Re: Need help with few questions

Good night!

$$\omega$$ is the imaginary cube root of unity.

All the numbers that are solution to:
$$x^3-1=0$$

So:
$$x^3-1=(x-1)(x^2+x+1)=0$$

Solving:
$$x=1$$

This is one solution. There are another 2:
$$x^2+x+1=0\\ \Delta=(1)^2-4(1)(1)=-3\\ x=\dfrac{-1\pm\sqrt{-3}}{2}\\ x=\dfrac{-1\pm i\sqrt{3}}{2}\\ x'=\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\\ x''=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}$$

If $$\omega=\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}$$, so
$$\omega^2=\left(\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\right)^2=\left(\dfrac{-1}{2}\right)^2+2\cdot\dfrac{-1}{2}\cdot i\dfrac{\sqrt{3}}{2}+\left(i\dfrac{\sqrt{3}}{2}\right)^2\\ \omega^2=\dfrac{1}{4}-i\dfrac{\sqrt{3}}{2}-\dfrac{3}{4}=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}$$

So, the solutions are:
1, $$\omega$$ and $$\omega^2$$

$$\omega^2+\omega+1=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}+\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}+1=-1+1=0$$

I hope to have helped!

Baltuilhe

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### Re: Need help with few questions

I don't know what to say. I am speechless. I am not able to understand a word you wrote. I am new to this, is there an easy way to solve it? Or you think I must Understand what you wrote.

Good night, thanx for trying to help. Hoping someone can help me with a easier way.

I have not studied about cube root of unity, so I have no idea about it. I am learning determinants only at this point, can this be solved using determinants.

I have studied Order 2 and Just started reading creamer's rule.

sunny

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### Re: Need help with few questions

I might have found an answer, please let me know if I am right.

After searching google I have came across a few help pages, that says its already proven that

$$W^{2}$$ + W+1 =0

$$W^{2}$$ + W

hence using above theorem we can say that my answer is -1

Am i thinking it correctly?

PS: I see that you have used symbol for unity, I can only see elected symbols in LaTeX Help, how can I find rest of the symbols?

sunny

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### Re: Need help with few questions

Solution to problem found so removing the post
sunny

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