fundamental theorem of algebra result

This leads to the following theorem:

Given a polynomialMathworldPlanetmathPlanetmathPlanetmath p(x)=anxn+an-1xn-1++a1x+a0 of degree n1 where ai, there are exactly n roots in to the equation p(x)=0 if we count multiple roots.

Proof The non-constant polynomial a1x-a0 has one root, x=a0/a1. 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 algebraMathworldPlanetmath a root zn. With polynomial division we find the unique polynomial q(x) such that p(x)=(x-zn)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, x4=0 has four roots, x1=x2=x3=x4=0.

Title fundamental theorem of algebra result
Canonical name FundamentalTheoremOfAlgebraResult
Date of creation 2013-03-22 14:22:01
Last modified on 2013-03-22 14:22:01
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Theorem
Classification msc 12D99
Classification msc 30A99