Fundamental Theorem Of AlgebraYou are currentlynot logged in Click here to log in 

The Fundamental Theorem of Algebra states that a polynomial equation of degree $n$ over the complex numbers has exactly $n$ (possibly duplicated) roots.
Actually the theorem states that a polynomial equation has at least one root. However, the factor theorem says that if $a$ is a root of $p(x)$ then $(xa)$ is a divisor of $p(x).$ Therefore we can write $p(x)=(xa)q(x)$ where $q(x)$ is a polynomial of lower degree, and the process can be repeated. Thus we can see that we must have exactly $n$ roots.
From wikipedia:
The fundamental theorem of algebra states that every nonconstant singlevariable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part.  The theorem is also stated as follows: every nonzero, singlevariable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. 

Last change to this page Full Page history Links to this page 
Edit this page (with sufficient authority) Change password 
Recent changes All pages Search 