State the possible number of imaginary zeros for the given function.
f(x)=-11x^4+8x^3+12x^2-1x-18

Question

State the possible number of imaginary zeros for the given function.
f(x)=-11x^4+8x^3+12x^2-1x-18

hero · Answer

Is there any particular method you're expected to use to solve this, because you could easily graph the expression to find the number of real zeroes. From that, you could determine how many imaginary zeroes there are.

daryan · Answer

any method is fine

hero · Answer

This is a 4th degree polynomial correct?

daryan · Answer

yes

hero · Answer

Since this is a fourth degree polynomial, we expect there to be four zeroes in total right?

hero · Answer

I mean, for example, if there were only two real zeroes, then there would be two imaginary zeroes. 

But if there were no real zeroes, then there would be four imaginary zeroes.

hero · Answer

The possibilities are that there will be either 0, 2, or 4 imaginary zeroes right?

daryan · Answer

yes

hero · Answer

So here is the graph: https://www.desmos.com/calculator/grbivhyvrl

hero · Answer

From that, how many real zeroes would you say there are?

daryan · Answer

I guess 1

hero · Answer

Can you explain how you arrived at your conclusion?

daryan · Answer

The graph intersects at zero only once from what I can see

hero · Answer

You do realize that the number of real zeroes corresponds with the number of times the graph crosses the x-axis right?

daryan · Answer

oh I did not know this

daryan · Answer

so it doesn't cross at all?

daryan · Answer

but see, the choices I am given are 2, 3, 4, or 5

hero · Answer

There's something else you should know in general about the relationship between real and imaginary zeroes.

In general, for a quartic equation:

If there are 4 real zeroes, then there are 0 imaginary zeroes. 
If there are 2 real zeroes, then there are 2 imaginary zeroes.
If there are 0 real zeroes, then there are 4 imaginary zeroes.