What are the possible number of positive real, negative real, and complex zeros of f(x) = -7x4 - 12x3 + 9x2 - 17x + 3?

Question

What are the possible number of positive real, negative real, and complex zeros of
f(x) = -7x4 - 12x3 + 9x2 - 17x + 3?

 

Choose one answer.
A. Positive Real: 3 or 1
Negative Real: 1
Complex: 2 or 0
 

B. Positive Real: 3 or 1
Negative Real: 2 or 0
Complex: 1
 

C. Positive Real: 1
Negative Real: 3 or 1
Complex: 2 or 0
 

D. Positive Real: 4, 2 or 0
Negative Real: 1
Complex: 0 or 1 or 3

Answer 1

First of all, complex roots always come in pairs -- because if a+bi is a root, then a-bi is also a root.  So there must be an even number of complex roots.  So answers B and D are not possible.

 

Then, use Descartes's rule and count the number of sign changes.  There are 3 sign changes.  So the number of positive roots is either 3 or 1.  So the answer is A.