Question

In this problem, you are given N 2D-points. You are asked to find a polynomial that passes through all those N points. The input has N+1 lines. The first line consists of a single positive integer N>2, which indicates the number of points and also the number of remaining lines. After that, each line consists of two integers, separated by a space, which are the x - and y-coordinates of a point respectively

Answer 1

To find a polynomial passing through N 2D-points, plot the points on a graph, determine their pattern, set up a system of equations and solve for the polynomial coefficients. The polynomial's degree will be at most N-1. It is important to understand each point's individual coordinates and how to use them to find the polynomial equation.

Step-by-step explanation:

To find a polynomial that passes through all given N 2D-points, the process involves several steps. First, graphs and tables are constructed to identify the dependence of y on x. By plotting given points like (1,5), (2,10), (3,7), and (4,14), one can see the pattern and dependence relationship. However, since we desire a polynomial, we typically look for the Lagrange or Newton's interpolating polynomial that fits through all points.

Since the question specifies that the polynomial must pass through all points, and if N is the number of points, the polynomial's degree will be at most N-1. The coordinates provided are used to set up a system of equations in which the polynomial coefficients are the unknowns. Solving these simultaneous equations yields the polynomial that exactly fits all the points.

Understanding that the best-fit line always passes through the sample mean of the x-values and y-values, denoted as (≡, ≡), is crucial. This is used when dealing with the linear case. However, for polynomials of higher degrees, it helps to know the individual x and y coordinates well. For instance, Y₂ and Y₁ are used to represent the y-values of a pair of points to calculate the slope if it was a straight line scenario, while X₂ and X₁ would be the x-values of those points. Yet for polynomials, we use these points with the polynomial equation to determine the coefficients.

When dealing with two-dimensional vectors, like the points given, it's easiest to work within a coordinate system with one horizontal axis (x) and one vertical axis (y) so we can project these points onto the axes. The polynomial is then derived through the use of these projections and relationships among the points.