Proving in math

1. Aug 27, 2004

Alkatran

Let's say I wanted to prove that, given n points, it takes a maximum of a (n-1)th degree polynomial to represent them all. How would I do it? My instinct is to just say because you need a max of (n-1) max/mins ...

2. Aug 27, 2004

matt grime

what does represent mean in this context? and are you sure you mean "maximum"

what degree one polynimial "represents" the two points 0 and 1. and point in what space? R, R^2, R^3...?

3. Aug 27, 2004

Alkatran

I mean that, given 1 point that your equation must touch, you need a 0th degree equation. Given 2 you need a 1st, etc...

For example, if you are given a set of points with 2 elements:
(a,b), (c,d)
You need a 1st degree equation, or line.
y = mx + e
The correct value of m and e will hit both points.

Similarly, if you have 3 points, you need a quadratic.

4. Aug 27, 2004

matt grime

In that case, given n points in the plane with distinct x coords, there exists a degree n-1 polynomial passing through them, since a degree n-1 poly has n coefficients and therefore you have a system of n linearly independent equations in n unknowns to solve.

you don't mean maximum at all since given n points then there is a polynomial of degree r=>n-1 passing through those points (again with distinct x values) which is unique when r=n-1.

5. Aug 27, 2004

TenaliRaman

Alkatran,
If u go through Lagrange Interpolation method, u would see how lagrange came up with an extremely simple way to do it!

6. Aug 27, 2004

Alkatran

I'm aware of how to solve the problem. My question was how do I prove that I will never need a 5th degree equation for 5 points?

7. Aug 27, 2004

TenaliRaman

Lagrange Interpolation Method works for any given n points.
Hence Proved!

8. Aug 27, 2004

matt grime

The proof that the equations formed by substituting in the n points are linearly independent is called the vandermonde determinant.