Linear Algebra - Polynomials Suppose that    $z_1 , . . . , z_{m+1}$    are distinct    elements    of  $\mathbb{R}$    and that $w_1, . . . , w_{m+1}$ ∈ $\mathbb{R}$. Prove that there exists a unique polynomial $p$ with degree $m$ such that
$p(z_j) = w_j$

Take the Lagrange interpolating polynomial.  This satisfies your requirements.  To see that it is unique, take the difference between it and any other polynomial.  It is a polynomial of order $m$ and so it can have at most $m$ roots.
