Solving a complex equation

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Homework Statement

Solve (z+1)^5 = z^5

None

The Attempt at a Solution

z^5 + 5z^4 + 10z^3 + 10z^2 + 5z + 1 = z^5
5z^4 + 10z^3 + 10z^2 + 5z + 1 = 0
5z^3(z + 2) + 5z(2z + 1) = -1

I'm not quite sure how to go about solving this. Expanding, canceling terms, and then factoring doesn't get me anywhere.

Homework Helper
Gold Member
Solve (z+1)^5 = z^5
It's clear that there are no real roots: consider the graphs of (x+1)^5 and x^5 for real x. Are you looking for an exact solution or a numerical approximation? For an exact solution, you'll have to solve a quartic equation, which can be done but it's fairly ugly.

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Homework Helper
Gold Member

Homework Statement

Solve (z+1)^5 = z^5

None

The Attempt at a Solution

z^5 + 5z^4 + 10z^3 + 10z^2 + 5z + 1 = z^5
5z^4 + 10z^3 + 10z^2 + 5z + 1 = 0
5z^3(z + 2) + 5z(2z + 1) = -1

I'm not quite sure how to go about solving this. Expanding, canceling terms, and then factoring doesn't get me anywhere.
Play with it.

5z4 + 10z3 + 10z2 + 5z + 1
=5(z4 + 2z3 + 2z2 + z) + 1

=5(z4 + 2z3 + z2 + z2 + z) + 1

=5( (z2 + z)2 + (z2 + z) ) + 1

=5 (z2 + z)2 + 5 (z2 + z) + 1​

Let u = z2 + z .

You have a quadratic equation in u .

Solve for u, then solve u = z2 + z for z.

Or just write your equation as $(\frac{z+1}{z})^5=1$. That tells you 1+1/z is a fifth root of unity. It's pretty easy from there.