# Polynomial

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A polynomial is an expression formed from one or more terms and constants by addition, each term of which should contain a variable raised to a non-negative integer power.

## Formal Definition

Let $R$ be a ring (such as the real numbers, rational numbers, or integers). Now let $n$ be a natural number and $a_0,a_1,\dots,a_n$ be elements of $R$. Then the function $p:R\rightarrow R$ defined by:

$p\left(x\right)=\sum_{k=0}^{n}a_{k}x^{k}=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{2}x^{2}+a_{1}x+a_{0}$

For all $x$ in $R$ is said to be a polynomial function in a single variable.

Moreover, if $a_n\ne 0$, we say that $p$ is an $n$th-degree polynomial.

We denote the set of polynomials as $R\left[x\right]$.

## Polynomial long division

$\frac{2x^4+2x^3-3x^3-2x+1}{x^2-1}$

Dividing a polynomial follows a process similar to long division.

\begin{align} & & & & +2x^2 & +2x & -1\\ \hline x^2 - 1 & | & 2x^4 & +2x^3 & -3x^2 & -2x & +1\\ & &-2x^4 & & +2x^2 & & \\ \hline & & & +2x^3 & -x^2 & -2x & +1\\ & & & -2x^3 & & +2x & \\ \hline & & & & -x^2 & & +1\\ & & & & -x^2 & & +1\\ \hline & & & & & & 0 \end{align}