Finding Solutions to a Complex Equation

[ƒ(x)]

Homework Statement

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

Homework Equations

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.

 

[jbunniii]

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.

 

[SammyS]

Homework Statement

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

Homework Equations

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.

Added in Edit:

See Dick's method in the next post. Sweet!

 

[Dick]

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.