Can someone help me find the exact solutions to the equation
?
I've used a grapher to find approximate solutions at the three places where it crosses the x-axis, and tried my best to work through Cardano's solution ( ) but I just can't finish it.
A big part of my problem here is that using the rational root theorem leads me to try , none of which work, so all three real solutions (it crosses three times, so no complex-imaginary roots) must be irrational. I recognize it obviously must be possible, but I'm not used to three irrational roots giving such a nice, simplistic equation. These three irrationals must multiply to be perfectly 3, add to be perfectly -2, and something else (what?) to be perfectly -9 (or +9? I can't remember).
Obviously I've put a good amount of thought into this problem; somebody please help me out!
Honestly, it's for my own understanding- I always believed that for polynomials with rational coefficients, any irrational solutions would come in conjugate pairs from quadratic formula use. You could use three cube root zeros to make a polynomial with a rational constant term, but I don't see how they would balance each other out for rational coefficients in other terms.
The short answer is, I'm interested to see the form the solutions take, and to use that to expand my own knowledge. Thanks for any help you can provide.
Okay, I'll admit: I was not expecting the solutions to include trig (though the and imply to me that this may have something to do with roots of unity, being equally spaced around a unit circle. True?)
Can you recommend a place to read up on this type of solution? The wikipedia article I linked to provides similar, but distinctly different solution methods. Incidentally, looking through my article, I'm noticing that the method I was trying "fails for the case of three real roots," which I wish I'd noticed earlier.
Thanks!