1.Using the factorization idea we have been working with, explain carefully how the truth of the statement "Every polynomial of degree at least one has one root" can provide a proof for the Fundamental Theorem of Algebra.

2.Consider the complex polynomialp(z) =z³zin a complex variablez. You found that there are two real

numbers wherep(z) = 3z²1is zero. For real inputs these were local extreme values and you can see a quadratic appoximation near these points. In the complex setting, what does the corresponding quadratic approximation predict regarding nearby inputs?

3.What does this example suggest is a good explanation of what happens near a critical point (zero of the derivative) for a polynomial? What if the critical point is a zero of the derivative of multiplicity2?3?