Math, asked by aditdwivedi336, 1 month ago

Please provide me the proof of fundamental theorem of Algebra.​
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Answers

Answered by nagrenikita769
5

♦ The Fundamental Theorem of Algebra is the statement

Every non constant polynomial has at least one zero in CC .

The most commonly found proof of this uses Liouvi-lle’s theorem:

Every bounded entire function is a constant.

The proof I am going to give can be found in any standard textbook on Complex Analysis.

Let, p (z) € C (z) , with deg f = n > 0 and suppose

p (z) ≠ 0 for all z € C. then,

f (z) = 1 / p(z).

is a entire, since it is a quotient of two analytic functions with a denominator never equal to 0 .

.The function f(z) is also bounded on C . To see this, note that as z→∞ , ∣p(z)∣ →∞ | p(z) | →∞ and so ∣ f(z)∣ →0 | f(z) |→0 . Consequently, there exists Rϵ R ϵ∈ R for which

R for which∣ f (z) <ϵ |f ( z) | < ϵ whenever |z| >| z| >Rϵ .

I don't know, you can take help from Goo-gle.

:(

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