Complex Numbers question

[Jean Valjean]

Hi everybody,

I have been given this two-part problem for homework, and I could complete the first part easily using either: 1) the Argand diagram to add the vectors geometrically, or 2) the formula for the sum of 5 terms of a geometric series.

For the second part though, I have no idea of where to even begin. I should mention that my teacher told me that this question is linked and that the answer to part a should in some way be used in part b...

The question goes :

Q) [MATH]z^5=1[/MATH] has roots [MATH]1, α, α^2, α^3, α^4[/MATH] where [MATH]α=cis(2π / 5)[/MATH]
a) Prove that [MATH]1+α+α^2+α^3+α^4=0[/MATH]
b) Solve [MATH]((z + 2)/(z - 1) )^5=1[/MATH], giving your answer in terms of α.

Thank you in advance and sorry if the formatting is not ideal.

 

[Deleted member 4993]

I was given this 2-part problem and although I could do part A geometrically and algebraically with no apparent problem, I am clueless as to where to start for part B. This question is supposed to use part A in part B, but I am not sure how.

Q) [MATH]z^5=1[/MATH] has roots [MATH]1, α, α^2, α^3, α^4[/MATH] where [MATH]α=cis(2π/5)[/MATH]
A) Prove that [MATH]1+α+α^2+α^3+α^4=0[/MATH]
B) Solve [MATH]((z+2)/(z-1))^5=1[/MATH], giving your answer in terms of [MATH]α[/MATH].

Thank you in advance.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem

 

[pka]

Look at this link.

 

[Dr.Peterson]

Hi everybody,

I have been given this two-part problem for homework, and I could complete the first part easily using either: 1) the Argand diagram to add the vectors geometrically, or 2) the formula for the sum of 5 terms of a geometric series.

For the second part though, I have no idea of where to even begin. I should mention that my teacher told me that this question is linked and that the answer to part a should in some way be used in part b...

The question goes :

Q) [MATH]z^5=1[/MATH] has roots [MATH]1, α, α^2, α^3, α^4[/MATH] where [MATH]α=cis(2π / 5)[/MATH]
a) Prove that [MATH]1+α+α^2+α^3+α^4=0[/MATH]
b) Solve [MATH]((z + 2)/(z - 1) )^5=1[/MATH], giving your answer in terms of α.

Thank you in advance and sorry if the formatting is not ideal.

Let w = (z+2)/(z-1). Does the equation in (b) then look related to that given above (a)?

I don't immediately see that you need to use the answer to (a) specifically; but sometimes you don't see what needs to be done until you've taken a few steps. So use alpha to solve your equation for w, and then for z. Then, look at your result and see whether (a) might be relevant. Maybe it will be useful in simplifying your answer ...

And, of course, show whatever work you've done, because you may have come closer to a useful answer than you realize.