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:

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

Backlinks

• A Programmers Introduction to Mathematics
  • polynomial
• Topic: Maths
  • polynomials
• Polynomial root
  • The root of a Polynomial $f: \\mathbb{R} -> \\mathbb{R}$ is a value $x$ for which $f(x) = 0$.

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1. A Programmers Introduction to Mathematics ↩