Polynomials

A polynomial of degree n is a function of the form
 p(x) = (Subscript[a, 0]) + (Subscript[a, 1])x + (Subscript[a, 2]) (x^2)+ ... +(Subscript[a, n  ])(x^n) where the ai are the coefficients.

p0[x_]:= 2;  (* constant *)
Plot[p0[x],{x,-2,2},PlotStyle-> Thick];

p1[x_]:= 3x +2; (* linear *)
Plot[p1[x],{x,-2,2},PlotStyle-> Thick];

Manipulate[Plot[a*x+b,{x,-2,2},PlotStyle->Thick],{a,-10,10},{b,-10,10}] ;(* explore linear polynomials *)

p2[x_]:= 2x^2 -2;
Plot[p2[x],{x,-2,2},PlotStyle-> Thick];

p3[x_]:= x^3;
Manipulate[Plot[p3[x]+a*x^2,{x,-4,4},PlotStyle-> Thick],{a,-5,5}];

Limits of polynomial fractions depend on the growth rates of numerator and denominator :

Limit[p2[x],x-> Infinity];
Limit[p3[x],x-> Infinity];
Limit[p2[x]/p3[x],x-> Infinity];
Plot[p2[x]/p3[x],{x,-20,20},PlotStyle->Thick];

Every polynomial of degree n has at most n  zeroes. Every odd degree one has at least one zero.

p5[x_]:=x^5+4x^2-1;
Plot[p5[x],{x,-2,2},PlotStyle-> Thick];
FindRoot[p5[x],{x,-.2}] ;(* initial guess: -.2 *)

A polynomial with prescribed zeroes.

p10[x_]:=Product[(x-i),{i,1,10}];
Plot[p10[x],{x,1,10},PlotStyle-> Thick];
Expand[p10[x]];

An even polynomial is symmetric wrt the y - axis. It has only even powers of x.

even[x_]:=3x^4-5x^2-1;
diffeven[x_]:=even[x]-even[-x];
Plot[{even[x],diffeven[x]},{x,-2,2},PlotStyle-> Thick];

An odd polynomial has only odd powers of x.

odd[x_]:=2x^5-3x^3-x;
diffodd[x_]:=odd[x]+odd[-x];
Plot[{odd[x],diffodd[x]},{x,-2,2},PlotStyle-> Thick];

A different even function,  not a polynomial .

diffcos[x_]:=Cos[x]-Cos[-x];
Plot[{Cos[x],diffcos[x]},{x,-10,10},PlotStyle-> Thick];

And an odd one

diffsin[x_]:=Sin[x]+Sin[-x];
Plot[{Sin[x],diffsin[x]},{x,-10,10},PlotStyle-> Thick];