# Polynomial¶

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}