Most recent change of RootsOfPolynomials

Edit made on December 08, 2008 by DerekCouzens at 10:07:15

Deleted text in red / Inserted text in green

A polynomial EQN:p(x) of degree n has n roots in the complex numbers. (see fundamental theorem of algebra).

If one of these roots is EQN:{\alpha} , then EQN:p({\alpha})=0

Furthermore, if EQN:p(x)=ax^n+bx^{n-1}+cx^{n-2}+...+px+q=0 has roots EQN:{\alpha}, EQN:{\beta}, EQN:{\gamma}, EQN:{\delta}, EQN:{\epsilon},...., EQN:{\omega} then

* EQN:\sum{\alpha}=-\frac{b}{a}
* EQN:\sum{\alpha}{\beta}=\frac{c}{a}
* EQN:\sum{\alpha}{\beta}{\gamma}=-\frac{d}{a}
* EQN:\sum{\alpha}{\beta}{\gamma}{\delta}=\frac{e}{a}
** etc...
* EQN:{\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=\pm\frac{q}{a}
*** If n is even then EQN:{\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=\frac{q}{a}
*** If n is odd then EQN:{\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=-\frac{q}{a}
Moreover, if one of the roots of EQN:p(x)=0 is EQN:z , where EQN:z is complex (i.e. EQN:z=x+iy and EQN:y is non-zero) and all the coefficients EQN:a,b,c...q are real then another root is the complex conjugate of EQN:z