Using Descartes' Rule and the Fundamental Theorem of Algebra to predict complex roots of polynomials

Question

I'm taking a pre-calculus course online, and teaching myself is turning out to be very difficult. Most recently, I have to do a written assignment as follows:

 

      How could you use Descartes' rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial?

      Your response must include:

       A summary of Descartes' rule and the Fundamental Theorem of Algebra. This must be in your own words.
      Two examples of the process. Provide two polynomials and predict the number of complex roots for each. You must explain how you found the number of complex roots for each.
       At least 100 words in complete sentences with appropriate grammar and spelling.
       Provide the source used for information (i.e. the website, the book etc.). Not providing your source may result in an Academic Integrity violation.

 

This is rather difficult for me as I have pretty much no earthly idea what any of this is. I'm good with math... or I thought I was until I started this course! Please, can anyone be of assistance? My grade is on the line here!

Answer 1

Descartes rule of signs

ax^3 +bx^2 +cx + d = 0

has a maximum number of positve real zeros equal to the number of sign changes

from a to be to c to d

if all are the same sign, there are no positve real zeros

if they alternate in sign, there is a maximum 3 positive real zeros

if one sign change, just one possible positive real zero

if two sign changes, then two possible positive real zeros

then to find the number of possible negative real zeros

substitute -x for x in the polynomial, then count the number of sign changes. the number of sign changes = the maximum number of negative real zeros

for example, if there is only 1 sign change for x, and no sign changes for -x, then there is 1 positive real zero and no real negative zeros. that leaves 2 imaginary zeros, they come in conjugate pairs

fundamental theorem of algebra gets more complicated and comes in several equivalent versions. See the wikipedia article.