How many roots does the polynomial z¹⁰⁰ − 50z³⁰ + 40z¹⁰ + 6z + 1 have in the open disc {z ∈ ℂ : |z|

Question

How many roots does the polynomial z¹⁰⁰ − 50z³⁰ + 40z¹⁰ + 6z + 1 have in the open disc {z ∈ ℂ : |z| < 1}?

Accepted Answer

C. On |z| = 1, |−50z³⁰| = 50 > 1 + 40 + 6 + 1 = 48 ≥ |z¹⁰⁰ + 40z¹⁰ + 6z + 1|, so by Rouché the polynomial has as many zeros inside as −50z³⁰: thirty.

Rouche dominant term