Simo K. Kivelä, 3 Nov 2007

Graph of a polynomial

If n+1 distinct points on the graph of a polynomial of degree n are given, the polynomial is uniquely determined. In the demonstration, the degree of the polynomial can be selected and then the necessary points are given. The graph corresponding the points is also plotted and it will change when points are moved. At the bottom, the expression of the polynomial and the coordinates of the points are given.

Manipulate[If[Length[p]≠n+1,p={},Null];p=PadRight[p,n+1,Table[{-n/2+k-1,1},{k,1,n+1}]];pol=Sum[a[k] x^k,{k,0,n}];yht=Table[(pol/.x->p[[k,1]])==p[[k,2]],{k,1,n+1}];kert=Table[a[k],{k,0,n}];pol2=pol/.First[Solve[yht,kert]];
ColumnForm[{Plot[pol2,{x,-5,5},PlotRange->{{-5,5},{-10,10}},ImageSize->400,AspectRatio->1.],pol2,N[p]}],{{p,{}},Locator},{{n,5,"degree"},{1,2,3,4,5,6,7,8,9,10}}]

Problems

1) What is maximum number of zeroes for a polynomial of degree n?

2) How many intersection points with a straight line may a polynomial of degree n have?

3) What can be said about the limits of a polynomial at ±∞?

4) How many local maximum and minimum points may a polynomial have?

5) How many inflection points may a polynomial have?