Question

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. Consider the complex polynomial p(x) = 2 – 2 in a complex variable 2. You found that there are two real numbers where p'(z) = 322 – 1 is 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? 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 multiplicity 2? 3?

Answer 1

Like in case of the real-valued functions, the point where derivative of the function is zero (also termed as a critical point)  is the point where the function is either not differentiable or the concavity of function changes.

Similarly, for the complex-valued function, the point where the derivative is zero is the point where the function is not analytic or the function changes its concavity in the argand plane.  

In the case of polynomials, a more clear physical significance can be given for the derivative of the polynomial. Suppose the derivative of a real-valued polynomial is alpha at some fixed point P on the graph of the polynomial. This alpha signifies the amount of stretch to the polynomial near point P. The order of derivative of the polynomial is the rate by which the point is stretched. Similarly, for complex-valued polynomial. If the alpha is imaginary then it shows the rotation instead of stretch. Now in our case, this alpha is zero it means there are some points on the graph of polynomial where no stretch or rotation has occurred.

Further, If the multiplicity of alpha is 2, the stretch is more flatty than the case of multiplicity 1. Also, as the multiplicity increases, the stretch gets more flatty.