Existence and Uniqueness

Theorem (Polynomial Existence and Uniqueness).  Given a set  n+1  of distinct nodes     (where whenever ).  There is a unique polynomial of degree    

        

that passes through the  n+1  points    

            for    .  

Proof.

    The existence of   is assured by using one of the standard methods of constructing a polynomial.  The Lagrange polynomial and Newton polynomial are two such constructions.

Mathematica Subroutine (Lagrange Polynomial). Compact object oriented programming.

Mathematica Subroutine (Newton Polynomial).

Exploration.

More Background

    We need to a result from algebra.

Theorem (The Fundamental Theorem of Algebra).   If    is a polynomial of degree with coefficients    that are complex numbers and then    has precisely   complex roots  .   

Proof.

Corollary.   Let    be a polynomial of degree with coefficients    that are complex numbers.  If there exists at distinct complex numbers     such that
            for    ,  
then
           for all complex numbers z.

Continuation of the Proof Polynomial Uniqueness Theorem.

    We are now ready to prove that the polynomial    is unique.  
Suppose that    is not unique and that there exists another polynomial    of degree which also passes through the  n+1  points  .  Form the difference polynomial

        

          
    
           
        
which is a polynomial of degree .  Furthermore, we have
    
        

           for    .  

Since at the  n+1  distinct values       we can conclude that

           for all complex numbers z.

Hence it follows that

           
    and
           for all complex numbers z.

Therefore    is unique.  

Exploration.

(c) John H. Mathews 2005