Question

Determine the polynomial $p(x)$ of degree 5 passing through the points $\left\{(0,0),\left(\frac{1}{2}, 0\right),(1,0),\left(\frac{3}{2}, 1\right),(2,0),\left(\frac{5}{2}, 0\right)\right\} .$ Determine the polynomials $L_{i}(x)$ for this set
of $x_{i}$ 's where
$$
L_{i}\left(x_{j}\right)=\left\{\begin{array}{ll}
0 & i \neq j \\
1 & i=j
\end{array}\right.
$$