Question

Let (x1, y1),(x2, y2),(x3, y3) be points with distinct x-values. Prove there exists a polynomial p(x) of degree at most 2 passing through these points. State and prove a similar result for four points?

Answer 1

Answer: we can use the folowing polynomial.

P(x) =
`(y1 (x - x2)(x -x3))/((x1 - x2)(x1-x3))` +
`(y2 (x - x1)(x -x3))/((x2 - x1)(x1-x3))` +
`(y3 (x - x2)(x -x1))/((x3 - x2)(x3-x1))`

you can see that P(x1) = y1

P(x2) = y2

P(x3) = y3

this is a Lagrange polynomial.