Your question could use some clarification. Why is the degree at most 3? Was that your choice?
What are the values 0,1,2,3,4,5...??
Given n points, you can create a polynomial of degree n-1 ...
Given a set of values how can one determine whether any polynomial (say p) of degree atmost 3 and having real coefficients exists such that for all i ?
E.g;
For the set of values:
0,1,2,3,4
there exists such a polynomial but for values:0,1,2,4,5 there isn't.
Is it related to some theorem?
Thanks.
Given any n (x,y) pairs there exist a polynomial of degree at most n-1 that passes through those points. It might happen that some of the n equations that are given by those values are not independent. Specifically, in order that there be a polynomial of degree 3, exactly 4 of the n equations must be independent, the other n- 4 equations linear combinations of those 4.
Hello, pranay!
Given a set of values how can one determine whether any polynomial.
say of degree at most 3 and having real coefficients exists
such that for all ?
E.g;
For the set of values: , there exists such a polynomial
but for values: , there isn't.
Is it related to some theorem?
This is a rehash of what HallsofIvy and TheChas said . . .
Given 2 points, there is always a line that passes through both points.
. . That is, there is a first-degree polynomial for them.
Given 3 points, there is always a parabola that passes through all the points.
. . That is, there is a second-degree polynomial for them.
There may or may not be a line (first-degree polynomial) for the 3 points.
Given 4 points, there ia always a cubic that passes through all the points.
. . That is, there is a third-degree polynomial for them.
There may or may not be a line or parabola for the 4 points.
Given 5 points, there is always a quartic that passes through all the points.
. . That is, there is a fourth-degree polynomial for them.
There may or may not be a line or parabola or cubic for the 5 points.
And the pattern should be clear . . .