We can construct the required polynomial $p(x)$ by using the Lagrange interpolation formula, which gives a polynomial of degree at most $n$ that passes through $n+1$ given points. Let $\ell_j(x)$ be the Lagrange basis polynomials defined as:

$$\ell_j(x) = \prod_{\substack{i=0 \ i \neq j}}^{n} \frac{x-x_i}{x_j-x_i}, \quad j=0,1,\ldots,n.$$

These basis polynomials have the property that $\ell_j(x_i) = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. That is, $\ell_j(x_i)$ equals $1$ if $i=j$ and equals $0$ if $i \neq j$.

Using these basis polynomials, we can construct the polynomial $p(x)$ as:

$$p(x) = \sum_{j=0}^{n} y_j \ell_j(x),$$

where $y_j$ is defined as:

1 & \text{if } j=j_0, \\ 0 & \text{if } j \neq j_0. \end{cases}$$ Thus, we have $p(x_{j_0}) = y_{j_0} = 1$ and $p(x_j) = y_j \ell_j(x_j) = 0$ for $j \neq j_0$. Moreover, the degree of $p(x)$ is at most $n$ since it is a linear combination of $n+1$ polynomials of degree at most $n$.

Gaussian quadrature is a numerical integration method that involves selecting specific quadrature points and weights for a given interval and a weight function. The goal is to approximate the integral of a function over that interval by a weighted sum of function evaluations at the quadrature points.

Gaussian quadrature using $n+1$ points involves selecting $n+1$ quadrature points $x_i$ and weights $w_i$ for the interval $[-1,1]$ and the weight function $w(x) = 1$. These quadrature points and weights are chosen so that the method is exact for polynomials of degree up to $2n+1$. Specifically, the quadrature rule is given by:

$$\int_{-1}^1 f(x) dx \approx \sum_{i=0}^n w_i f(x_i)$$

where the quadrature points $x_i$ and weights $w_i$ are determined by the roots and weights of the $n+1$-th order Legendre polynomial $P_{n+1}(x)$.

To show that this method is exact for polynomials of degree up to $2n+1$, we need to show that for any polynomial $p(x)$ of degree $k \leq 2n+1$, the quadrature rule above gives the exact result for the integral of $p(x)$ over $[-1,1]$.

Let $p(x)$ be a polynomial of degree $k \leq 2n+1$. We can write $p(x)$ in terms of the Legendre polynomials as:

$$p(x) = \sum_{i=0}^k c_i P_i(x)$$

where $c_i$ are the coefficients of $p(x)$. Note that since $k \leq 2n+1$, this sum involves at most $n+1$ terms.

Then, the integral of $p(x)$ over $[-1,1]$ can be written as:

$$\int_{-1}^1 p(x) dx = \sum_{i=0}^k c_i \int_{-1}^1 P_i(x) dx$$

Now, since the Legendre polynomials are orthogonal with respect to the weight function $w(x) = 1$ over $[-1,1]$, we have:

$$\int_{-1}^1 P_i(x) P_j(x) dx = 0 \quad \text{if } i \neq j$$

and

$$\int_{-1}^1 P_i^2(x) dx = \frac{2}{2i+1} \quad \text{for } i \geq 0$$

Using these properties, we can simplify the integral of $p(x)$ as:

$$\int_{-1}^1 p(x) dx = \sum_{i=0}^k c_i \int_{-1}^1 P_i(x) dx = \sum_{i=0}^k c_i \frac{2}{2i+1} \delta_{i,k}$$

where $\delta_{i,k}$ is the Kronecker delta function, which is $1$ if $i=k$ and $0$ otherwise.

Thus, the integral of $p(x)$ can be written as:

$$\int_{-1}^1 p(x) dx = \frac{2}{2k+1} c_k$$

Now, using the quadrature rule for Gaussian quadrature with $n+1$ points

The Legendre polynomials $P_n(x)$ are a family of orthogonal polynomials over the interval $[-1,1]$ with respect to the weight function $w(x) = 1$. They have many important properties and applications in mathematics and physics, including in numerical analysis, approximation theory, and quantum mechanics. The first few Legendre polynomials are given by:

$$P_0(x) = 1$$

$$P_1(x) = x$$

$$P_2(x) = \frac{1}{2}(3x^2-1)$$

$$P_3(x) = \frac{1}{2}(5x^3-3x)$$

$$P_4(x) = \frac{1}{8}(35x^4-30x^2+3)$$

$$P_5(x) = \frac{1}{8}(63x^5-70x^3+15x)$$

$$P_6(x) = \frac{1}{16}(231x^6-315x^4+105x^2-5)$$

$$P_7(x) = \frac{1}{16}(429x^7-693x^5+315x^3-35x)$$

$$P_8(x) = \frac{1}{128}(6435x^8-12012x^6+6930x^4-1260x^2+35)$$

$$P_9(x) = \frac{1}{128}(12155x^9-25740x^7+18018x^5-4620x^3+315x)$$

$$P_{10}(x) = \frac{1}{256}(46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)$$

The Legendre polynomials have several important properties that determine their properties and allow them to be calculated efficiently. Some of these properties include:

1. Orthogonality: $P_n(x)$ is orthogonal to $P_m(x)$ with respect to the weight function $w(x) = 1$ over the interval $[-1,1]$, i.e. $\int_{-1}^1 P_n(x) P_m(x) dx = 0$ if $n \neq m$.
2. Recurrence relation: The Legendre polynomials satisfy a recurrence relation of the form $(n+1)P_{n+1}(x) = (2n+1)x P_n(x) – n P_{n-1}(x)$ for $n \geq 1$, with $P_0(x) = 1$ and $P_1(x) = x$.
3. Explicit formula: The Legendre polynomials can be calculated explicitly using the Rodrigues formula, which states that $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n$.
4. Zeros: The Legendre polynomials have $n$ real zeros in the interval $[-1,1]$, which can be used as quadrature points in numerical integration methods such as Gaussian quadrature.
5. Symmetry: The Legendre polynomials are even or odd functions of $x$ depending on whether $n$ is even or odd, respectively.
6. Norm: The Legendre polynomials are normalized so that $P_n(1) = 1$.

These properties make the Legendre polynomials useful for a wide range of applications in mathematics and physics.