Polynomial

Author

Eric W. Weisstein
January 14, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Polynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Polynomial.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Typesetting

Options[TraditionalForm]

poly=Expand[(x+1)^7];

poly//TraditionalForm

PolynomialForm[poly,TraditionalOrder->True]

PolynomialForm[Expand[(x+1)^7],TraditionalOrder->True]//TraditionalForm

x y^2+x^2 y+1//TraditionalForm

x y^2+x^2 y+1//TraditionalForm[#,ParameterVariables:>{y}]&

Example

Polynomial[{a,b,c},x]

Expand[%]

Products

Sum[a[n]x^n,{n,2,0,-1}]Sum[b[n]x^n,{n,1,0,-1}]

Collect[%,x]

Long Division

(((x^4)+x+1)/(x+1))==Together[(x^3)-(x^2)+x+(1/(x+1))]

Nesting Identity

For any integer polynomial P(x), P(x)-x divides P(P(x))-x.

RandomIntegerPolynomial[n_,x_]:=Polynomial[Table[Random[Integer,{-10,10}],{Random[Integer,{1,n}]}],x]

RandomRationalPolynomial[n_,x_]:=Polynomial[Table[Random[Integer,{-10,10}]/Random[Integer,{1,10}],{Random[Integer,{1,n}]}],x]

Table[Cancel[(Nest[#,x,2]-x)/(#[x]-x)&[Function[x,Evaluate[RandomIntegerPolynomial[5,x]]]]],{10}]//Expand

Denominator/@Table[Cancel[(Nest[#,x,2]-x)/(#[x]-x)&[Function[x,Evaluate[RandomIntegerPolynomial[10,x]]]]],{10}]//Union

Table[Cancel[(Nest[#,x,2]-x)/(#[x]-x)&[Function[x,Evaluate[RandomRationalPolynomial[5,x]]]]],{10}]

Denominator/@Table[Cancel[(Nest[#,x,2]-x)/(#[x]-x)&[Function[x,Evaluate[RandomRationalPolynomial[10,x]]]]],{10}]//Union

Cancel[(Nest[#,x,2]-x)/(#[x]-x)&[Function[x,1+x/2]]]