Expert: Richard J. Raridon - 4/24/2010

Question
I attend an online school so I'm teaching myself all of these concepts (because my family isn't all that great at math) and generally, I do alright...I can get it most of the time. However, sometimes concepts just don't "click" right away and I have to search for someone to walk me through a problem. Now is one of those times.

I'm supposed to find the possible number of imaginary zeros of the function f(x) = 7x^3 - x^2 + 10x -4. I tried using Descartes' Rule of Signs and came up with 3 sign changes in total and no negative real zeros. So that would mean that there are 3 positive real zeros and no imaginary zeros, 2 positive real zeros and 1 imaginary zero, 1 positive real zero and 2 imaginary zeros or 3 imaginary zeros. I just don't know where to go from there...

Like I said...i don't understand how to figure possible imaginary zeros all that well so I'm not even sure if what i have so far is correct.

Could you walk me through this by any chance?

Answer
When I took algebra 60 years ago they didn't teach sign changes.  However, I know, and you should know, that imaginary zeros always come in pairs.  
Since f(0) = -4 and f(1) = 12, there's one zero between 0 and 1 and there are no negative zeros so I'm assuming there is one positive real zero and two imaginary zeros.