Real and complex Roots of A cubic equation

[mathsTKK]

Homework Statement

Find the roots of y-y^3=1

Homework Equations

The Attempt at a Solution

Factor theorem doesn't help in this equation. Found from Wikipedia, a very complex formula is needed to solve cubic eq. Can someone show me the steps to find the root?

Thank you ^^

 

[HallsofIvy]

That is the same as y^3- y+ 1= 0. By the "rational root theorem" the only possible rational roots are 1 or -1 and it is easy to see that neither of those satisfy the equation. That is, this equation has only irrational roots. The only way to get exact roots is to use that cubic formula that you cite.

 

[I like Serena]

Hi mathsTKK! Welcome to PF!

Since your polynomial does not contain a second order term, you can apply Cardano's method directly. See:
http://en.wikipedia.org/wiki/Cubic_function#Cardano.27s_method

If you want to solve your equation algebraically, this is basically the only way to go.
I won't be able to explain it better than it is already explained in the article.

Of course if you're only interested in a solution, you can always visit http://www.wolframalpha.com.