Question

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

Answer 1

I would start by graphing a bunch of polynomials and noticing when you find ones with complex solutions. The fundamental theorem tells us there will be the highest degree number of solutions at which a term in the polynomial is taken too.

Answer 2

Start small.

y = x-2

How many sign changes can there possible be with a linear function?

Move up to quadratic. Make a catalogue.

What other properties can you discern from your catalogue?

Answer 3

im confused

Answer 4

+x+2 -- No sign change
+x-2 -- One sign change
-x-2 -- No sign change
-x+2 -- One sign change

How many sign changes can there possibly be with a linear function? Maximum of 1

How many sign changes can there possibly be with a quadratic function? Maximum of 2

How many sign changes can there possibly be with a cubic function? ??

This establishes the total cumber of roots. This also establishes SOME information concerning Positive Real Roots.