Can someone explain how to find how many solutions a polynomial has over the complex number system?

Question

anonymous · Answer

Specifically for the polynomial$$7x ^{5}-33x ^{4}-4x ^{2}+3x+52=0$$

hero · Answer

First figure out how many real solutions it has. If you figure out all the real solutions, say for example, you figured out it had three real solutions, then you would know that 2 complex solutions remain.

anonymous · Answer

Hmmm, is the real solutions based on how many times the signs change?

anonymous · Answer

If so, it would be one real solution... right?

hero · Answer

How do you figure?

anonymous · Answer

Oh wait, two, because it goes positive, negative, negative, positive, positive

hero · Answer

Well, it doesn't work exactly in that manner. That method gives the highest possible number of solutions. There could be 2 or zero. The quickest way to figure this out is by graphing the polynomial and observing how many times the graph hits the x axis.

anonymous · Answer

i have a graphing calculator, I'll graph it real quick

hero · Answer

Also, you may want to review this: http://www.purplemath.com/modules/drofsign.htm

anonymous · Answer

It passes through the x axis three times

hero · Answer

Yup, so three real solutions and how many complex?

anonymous · Answer

two complex solutions

anonymous · Answer

thank you!

hero · Answer

Now trying to figure it out algebraically is an entirely different beast altogether.