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[parent] fundamental theorem of algebra result (Theorem)

This leads to the following theorem:

Given a polynomial $ p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0 $ of degree $ n\geq 1$ where $ a_i\in \mathbb{C}$, there are exactly $ n$ roots in $ \mathbb{C}$ to the equation $ p(x)=0$ if we count multiple roots.

Proof The non-constant polynomial $ a_1x-a_0$ has one root, $ x=a_0/a_1$. Next, assume that a polynomial of degree $ n-1$ has $ n-1$ roots.

The polynomial of degree $ n$ has then by the fundamental theorem of algebra a root $ z_n$. With polynomial division we find the unique polynomial $ q(x)$ such that $ p(x)=(x-z_n)q(x)$. The original equation has then $ 1 + (n-1)=n $ roots. By induction, every non-constant polynomial of degree $ n$ has exactly $ n$ roots.

For example, $ x^4=0$ has four roots, $ x_1=x_2=x_3=x_4=0$.



"fundamental theorem of algebra result" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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multiplicity (Definition) by pahio
irreducible polynomial (Definition) by pahio
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Cross-references: induction, division, fundamental theorem of algebra, proof, multiple roots, equation, roots, degree, polynomial

This is version 4 of fundamental theorem of algebra result, born on 2004-05-11, modified 2004-11-05.
Object id is 5851, canonical name is FundamentalTheoremOfAlgebraResult.
Accessed 3355 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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grammatical disagreement by rspuzio on 2004-09-13 20:26:49
The "is" needds to be plularized to "are" in the statement of the theorem.
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