polynomial interpolation
Suppose a sensor logged a temperature only once an hour, but you need a smooth estimate for any moment in between. You want a single, simple curve that passes EXACTLY through every recorded dot, so you can read off values at times you never measured. Polynomial interpolation is the most basic way to build that curve: you fit a polynomial — a sum of powers like a_0 + a_1 x + a_2 x^2 + ... — through the data points.
The central fact is a uniqueness theorem: given n + 1 points with DISTINCT x-values, there is one and only one polynomial of degree at most n that passes through all of them. Two points fix a straight line (degree 1), three points fix a parabola (degree 2), and so on. You can hunt for the coefficients in several equivalent ways — solving a linear system (the Vandermonde matrix), writing the answer directly as a weighted sum of data values (the Lagrange form), or building it up point by point (the Newton divided-difference form). All three produce the SAME polynomial; they differ only in cost, numerical stability, and how easily you can add a new point.
Interpolation is the foundation under quadrature (integrate the interpolant instead of the unknown function), numerical differentiation, and ODE/PDE solvers, because replacing an awkward function by a polynomial makes it easy to integrate, differentiate, and evaluate. The crucial honesty: passing through the points does NOT guarantee a faithful curve between them. A single high-degree polynomial through many EQUISPACED points can oscillate wildly and even diverge (Runge's phenomenon), so in practice one either chooses special nodes (Chebyshev nodes) or abandons one global polynomial for many low-degree pieces (splines).
Fit a polynomial through (0, 1), (1, 2), (2, 5). Three points force a degree-2 curve. Solving gives p(x) = x^2 + 1: check p(0) = 1, p(1) = 2, p(2) = 5. Now you can estimate p(1.5) = 3.25 — a value at a point you never sampled.
Three points pin down exactly one parabola.
Uniqueness needs DISTINCT x-values; two points sharing an x but different y have no interpolating polynomial. And 'unique polynomial of degree at most n' does not mean degree exactly n — collinear points give a lower-degree answer.