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A polynomial is a function with n coefficients, where n is known as the degree. A single-variable polynomial f has the form

f(x) = a_0 + a_{1}x^1 + \dots + a_{n}x^n

Here a_n are the coefficients of the polynomial.

Example Theorem1

For any integer n \geq 0 and any list of n + 1 points (x_0,y_0),(x_1,y_1)\dots,(x_{n+1},y_{n+1}) in \mathbb{R}^2 where x_0 < x_1 < \dots < x_n there exists a unique degree n polynomial p(x) such that p(x_i) = y_i for all i.

Lets try it out.

Where n = 0 our list would be (x, y) - the easy solution here is f(x) = y

Where n = 1 we have (x_1, y_1), (x_2, y_2). In order to have f(x_1) = y_1, the first coefficient must be y_1 the second coefficient has to evaluate to 0, the identity for addition. The second coefficient is multiplied by x to the power of 1, so if the second coefficient is 0 this will happen

With the coefficients y_1, 0 $$ (f(x_1) = y_1 + 0*x_1) = (f(x_1) = y_1) $$

But does this work for f(x_2) = y_2?

f(x_2) = y_1 + 0*x_2

This will still resolve to y_1 - which doesn't fit. So we need our coefficients to be more complex. For each x, one coefficient should be the corresponding y and one must be 0.

As I have a textbook, I can jump ahead a little - I'd like to explore how to figure this out more deeply at a later date, but for now:

f(x) = y_1 \frac{x - x_2}{x_1 - x_2} + y_2 \frac{x - x_1}{x_2 - x_1}