Polynomial

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 $a_{n}\neq 0$, 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.

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

See also

• Factoring