02 - Uniqueness of Interpolating Polynomial

Uniqueness of Interpolating Polynomial

Interpolating Polynomial of Degree $\le n$ is unique

Proof

Consider two interpolating polynomials $p_n,q_n\in {\Pi }_n$

Consider difference $d_n=p_n-q_n\in {\Pi }_n$ then

$$d_n(x_k)=0\text{ for }k=0,1,\cdots ,n$$

As $d_n$ is a polynomial of degree at most $n$ but it has at least $n+1$ roots

By the Fundamental Theorem of Algebra then

$$d_n\equiv 0⟹p_n=q_n$$