I have given the following nonconstant complex polynomial h ( x ) = x n </ms

Dale Tate 2022-06-27 Answered
I have given the following nonconstant complex polynomial h ( x ) = x n + a n 1 x n 1 + + a 1 x + a 0 . In the lecture our Prof. told us that using the minimal principle one could find z C such that h ( z ) = 0
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Answers (1)

Hadley Cunningham
Answered 2022-06-28 Author has 20 answers
Suppose h ( z ) is a polynomial of degree at least 1 and h ( z ) has no roots.
Consider g ( z ) = 1 h ( z ) . If h has no roots, g is analytic.
| g ( 0 ) | > 0 (or else we have found a root and have a contradiction.)
Over the domain | z | R the maximum modulus theorem says the maximum of | g ( z ) | must lie on the circle | z | = R
But if as |z| gets to be large, | h ( z ) | goes to infinity.
Hence for large enough R, | z | = R implies 0 < | g ( z ) | < | g ( 0 ) | which is a contradiction.

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