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

ax^3 + bx^2 -cx + d = 0 assume a, b, c and d>0

the polynomial equation has 3 roots, according to Gauss' Fundamental Theorem of Algebra

Number of roots = the exponent of the highest degree term = 3 for ax^3

Descartes' rule of signs means the equation has maximum 2 positive real solutions

= number of sign changes

it also says maximum 1 negative real solution

since imaginary solutions come in conjugate pairs

there are two possibilities

either there are 2 positive real, 1 negative real, and zero imaginary solutions or roots

Or, there are 2 imaginary and 1 negative real solution.

if the graph of the polynomial function is tangent to the x axis, there will be 1 solution which repeats, counting the repetition as another root, means 2 roots which are identical. That makes the Fundamental Theorem of Algebra true, that there are as many roots as the exponent of the highest degree term