Complex Numbers

[Muddyakka]

Find the 4 roots to: (z is an element of the complex field)
z⁵ = (z+1)⁵, in the form of a + bicot(theta)

I have just expanded the right side to cancel the z⁵ term but I am stuck at the polynomial after. Co-efficients are 5 10 10 5 1, in decreasing order.

I have tried grouping but there is a lack of symmetry (5, 5 differ by 3 degrees while 10 10 differ by 1), while i can factor out a z with a degree of 2 it wont cancel with 10's.

I have also tried applying Vietta's theorem but with 4 possible roots those simultaneous equations are nasty. (Even though conjugate root theorem holds)

Finally I tried letting z be the general solution of a + ibcot(theta) but even that doesnt simplify nicely.

(EDIT: I forgot to mention that I also tried using cis form as the cot(theta) implies cis. But I end with cos and sin of various degrees or with various thetas (4theta, 3theta, as an applocation of De Moivres))

 

[pka]

Find the 4 roots to: (z is an element of the complex field)
z⁵ = (z+1)⁵, in the form of a + bicot(theta)

Have a look at this LINK.

 

[Cubist]

Try the substitution u = z + 1/2

EDIT: This leads to \( 80u^4 + 40u^2 + 1 = 0 \)

 

[lex]

Clearly z≠0
[MATH](1+1/z)^5=1[/MATH][MATH]1+1/z=e^{2k \pi i/5}[/MATH] for k=1 to 4. (Clearly 1+1/z≠1)
You should be able to find the 4 values of z.

 

[Muddyakka]

Thank you all for the elegant solutions!