Complex Analysis week 3Preben Alsholm, September 16 2008
<Text-field style="Heading 1" layout="Heading 1">Packages</Text-field>restart;libname:="F:/DMat/libDMat",libname;with(DMat):with(plots):with(RandomTools):
<Text-field style="Heading 1" layout="Heading 1">The division algorithm for polynomials</Text-field>The procedure PolyDiv is found in the DMat package.p:=x^5-6*x^3+6*x^2+5*x+12;q:=x^2+2*x+3;PolyDiv(p,q,x);p:=RandomTools[Generate](polynom(integer(range=-10..10), x, degree=7));q:=RandomTools[Generate](polynom(integer(range=-10..10), x, degree=2));PolyDiv(p,q,x);quo(p,q,x,'r');r;
<Text-field style="Heading 1" layout="Heading 1">The Fundamental Theorem of Algebra: An Illustration </Text-field>Any non-constant polynomial with complex coefficients has at least one root within the complex numbers.An immediate corollary is: Any non-constant polynomial can be written as a product of factors of degree one within C.A random polynomial of degree 6 with integral coefficients between -9 and 9:p:=Generate(polynom(integer(range=-9..9),z,degree=6));The roots (numerically):fsolve(p=0,z,complex);factor(p,complex);Throw away som decimals:evalf[3](%);
<Text-field style="Heading 1" layout="Heading 1">A randomly chosen polynomial of high degree</Text-field>Degree 100:p:=Generate(polynom(integer(range=-99..99),z,degree=100));r:=fsolve(p=0,z,complex):complexplot([r],style=point,symbol=circle,scaling=constrained);There seems to be somewhat of a pattern.