Please solve step by step and Don't use ChatGPT and Do not use old chegg answers..

Topics covered: polynomials. 2.1. (Polynomial interpolation) Suppose S={(x0,y0),(xn,yn)} is a set of n points of RR. 1 The objective of polynomial interpolation is to find a polynomial P(x) (the interpolating polynomial) of lowest possible degree such that the graph of the function y=P(x) passes through all of the points of S. A. Prove that if all of the first coordinates xi of elements in S are distinct then there exists a unique polynomial P(x) of degree n that is the interpolating polynomial for S. B. Verify that, in the setting of the previous subsection, the Lagrange interpolating polynomial L(x) given by the formula L(x)=i=0nyiLi(x),Li(x)=k=ixixkxxk is the interpolating polynomial for S. 2.2. Recall that p>1 is a prime number if and only if (p1)!1(modp). (Wilson's theorem, problem 4.5 from last year). Explain how this statement can be seen as a consequence of Vieta's formulas. 2.3. Prove that cos52=451 by using complex numbers and considering the set of all roots of the polynomial f(x)=x51. 2.4. Let K[x1,,xn] be the K-algebra of polynomials in n variables and for a monomial x1k3xnkn let the degree deg(x1k1xnkn) be k1++kn. A. How many distinct monomials x1k2xnkn of a given degree d are there? B. Find the dimension of the K-vector space K[x1,,xn]d of polynomials of degree less than or equal to d. 2.5. The lexicographical order on the set Nn of n-tuples of natural numbers (the naturals are equipped with the standard total order) is defined as follows; we say that (k1,,kn)(l1,,ln) if k1