Strange Analysis Polynomial Question (Lagrange Interpolation?)

Question

We have $0\leq x_0\lt x_1\lt\,...\,\lt x_N\leq1$, where all $N+1$ distinct points are in $[0,1]$ and I must show: If $p$ is a polynomial with degree $\leq N$, and $$Q_k(t)=\prod_{\substack{0\leq i\leq N}\\ \:\:\:i\neq k}\frac{t-x_i}{x_k-x_i}$$ then $$p=\sum_{k=0}^Np(x_k)Q_k.$$ I have honestly no idea where to start. I've done some research and can see this relates to Lagrange polynomials, however, that is well beyond the scope of the course I am on - I believe I'm meant to use basic analysis methods / ingenuity to prove this,

Answer 1

This relies on two observations.

$1.)$ The function defined by $f = \sum_{k=0}^Np(x_k)Q_k$ satisfies that $f(x_i) = p(x_i)$ for $0 \leq i \leq N$. (This is because $Q_k(x_i) = \delta_{ik}$

$2.)$ If two polynomials of degree $N$ agree on $N+1$ points are the same.

From this you can conclude $$ \sum_{k=0}^Np(x_k)Q_k = p$$

To prove $2$, suppose $f,g$ were two polynomials of degree $N$ for which $f(x_i)= g(x_i)$. Then $f-g$ is a polynomial of degree $\leq N$ with $N+1$ roots: $x_0, \dots x_N$.

Not possible unless $f-g = 0$ or $f=g$.