Information on Student Research Return

 A polynomial of degree $n$ is a function $p(x)$ of the form $$p(x) = a_0 + a_1 x + \ldots + a_n x^n,$$ where the coefficients $a_1$, $a_2, \ldots a_n \in \mathbb{R}$ and $a_n \neq 0$. This is the canonical form of the polynomial with which we are familiar and corresponds to the Taylor series expansion for the function $p(x)$ about the point zero, i.e., $$p(x) = p(0) + \frac{p^{\prime} (0) x}{1!} + \frac{p^{\prime \prime} (0) x^2}{2!} + \ldots + \frac{p^{(n)} (0) x^n}{n!},$$ where $a_i = p^{(i)}(0)/i!$, $i = 0,1,\ldots,n$. Several alternative representations of the polynomial $p(x)$ are useful, of which the Lagrange form: $p(x) = a_0 l_0(x) + a_1 l_1(x) + \ldots + a_n l_n (x),$ where $$l_k(x) = \prod_{\stackrel{i=0}{i\neq k}}^n \frac{x-x_i}{x_k - x_i}, \qquad k = 0,1, \ldots n.$$ will be of interest.