# Thread: Finding whether a polynomial exists

1. ## Finding whether a polynomial exists

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 $p(i) = x_i$ 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.

2. 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 ...

3. 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.

4. Hello, pranay!

Given a set of values how can one determine whether any polynomial.
say $p(x)$ of degree at most 3 and having real coefficients exists
such that $p(i) = x_i$ 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?

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 . . .