Complex Numbers / Polynomials Q (1 Viewer)

[mreditor16]

if I have w as a complex root of z^5 = 1, then how would you express the conjugate of (w + w^4) in terms of positive powers of w


(Asking on behalf of a friend)

 

[InteGrand]

 

[mreditor16]

Let me rephrase my original Q -



I am stuck on part ii)

 

[InteGrand]

By the conjugate root theorem, the other root is (i.e. double root).

 

[Sy123]

uh nah :/

here's the original Q - I'm stuck on part ii)



Do you wanna reply on new thread? Feel bad for posting this on NS.

I have a feeling that there is a typo in the question and that

Because as it is being asked now, the answers are trivial, and they don't even aid in getting part (iv)

 

[InteGrand]

Part (iii) asks something about a – what's a? Do they mean find the values of b and c?

 

[mreditor16]

By the conjugate root theorem, the other root is (i.e. double root).

Noob Q here (because I've forgotten my shit) - how is conjugate of w equal to w^4 ?

 

[mreditor16]

Part (iii) asks something about a – what's a? Do they mean find the values of b and c?

I assume so

 

[Sy123]

Noob Q here (because I've forgotten my shit) - how is conjugate of w equal to w^4 ?

I am willing to bet that there is a typo in what is supposed to be

makes more sense with part (iv)

Anyway, for a complex number z, its conjugate is such that z*conj(z) = real number

So, since thus w and w^4 are conjugates

 

[mreditor16]

I am willing to bet that there is a typo in what is supposed to be

makes more sense with part (iv)

Anyway, for a complex number z, its conjugate is such that z*conj(z) = real number

So, since thus w and w^4 are conjugates

Omg I feel so stupid - I still don't get it.

 

[Sy123]

Omg I feel so stupid - I still don't get it.

So remember that w is a root of z^5 = 1

This means that w^5 = 1, right?

Since w^5 = 1

and w^4*w = w^5 (since when we multiply them the powers add)

thus w^4 * w = 1

So what we have here is 2 complex numbers, that multiplied together get a real number (and both complex numbers have the same modulus), therefore they are conjugates of each other

meaning

 

[seanieg89]

I am willing to bet that there is a typo in what is supposed to be

makes more sense with part (iv)

Anyway, for a complex number z, its conjugate is such that z*conj(z) = real number

So, since thus w and w^4 are conjugates

Definitely. The question doesn't make sense as is, the conjugate root theorem tells us nothing if one of the roots is real. Beta can be absolutely anything.

 

[mreditor16]

So remember that w is a root of z^5 = 1

This means that w^5 = 1, right?

Since w^5 = 1

and w^4*w = w^5 (since when we multiply them the powers add)

thus w^4 * w = 1

So what we have here is 2 complex numbers, that multiplied together get a real number (and both complex numbers have the same modulus), therefore they are conjugates of each other

meaning

How can we just make that assumption? What if there is another complex number multiplied by w which gives a (different) real number?

 

[InteGrand]

How can we just make that assumption? What if there is another complex number multiplied by w which gives a (different) real number?

for some such that for .

, since differs from by a multiple of .

 

[Sy123]

How can we just make that assumption? What if there is another complex number multiplied by w which gives a (different) real number?

Well it comes directly from the definition

Two conjugates of each other are defined as:







So, if I have 2 complex numbers of the same modulus, and when multiplied they become a real number, it means the arguments of the complex numbers add up to zero, this means their arguments are opposite in sign.

And by definition 2 complex numbers of equal modulus and arguments opposite in sign are conjugates of each other

 

[seanieg89]

How can we just make that assumption? What if there is another complex number multiplied by w which gives a (different) real number?

Remember the rule



This implies for nonzero z:



In particular, inverting is the same thing as taking conjugates on the unit circle.

For example



if w is a 5-th root of unity.

 

[mreditor16]

I get it guys. Thank you so much!!!