Standard form of a Polynomial in a single variable

General form

$a_{n}\,x^{n}$ $+$ $a_{n-1}\,x^{n-1}$ $+$ $a_{n-2}\,x^{n-2}$ $+$ $\cdots$ $+$ $a_{2}\,x^2$ $+$ $a_{1}\,x$ $+$ $a_{0}$

Introduction

Let $x$ be a variable or an indeterminate value. Suppose, $a_{0},$ $a_{1},$ $a_{2},$ $\cdots$ $a_{n-2},$ $a_{n-1}$ and $a_{n}$ are the constants. The addition of the products of a constant and a non-negative integer exponentiation of a variable $x$ forms an expression as follows.

$a_{n}\,x^{n}$ $+$ $a_{n-1}\,x^{n-1}$ $+$ $a_{n-2}\,x^{n-2}$ $+$ $\cdots$ $+$ $a_{2}\,x^2$ $+$ $a_{1}\,x$ $+$ $a_{0}$

This mathematical expression is called the standard form or general form a polynomial in one variable. It is also written as follows in ascending order

$a_{0}$ $+$ $a_{1}\,x$ $+$ $a_{2}\,x^2$ $+$ $\cdots$ $+$ $a_{n-2}\,x^{n-2}$ $+$ $a_{n-1}\,x^{n-1}$ $+$ $a_{n}\,x^{n}$

Let’s learn some more about the general form of polynomial in single variable.

Terms

The single variable polynomial consists of several expressions and a plus sign connects every two expressions in algebraic form. Each expression is called a term of the polynomial.

$a_{n}\,x^{n},$ $a_{n-1}\,x^{n-1},$ $a_{n-2}\,x^{n-2},$ $\cdots$ $a_{2}\,x^2,$ $a_{1}\,x$ and $a_{0}$ are the expressions in the polynomial and all expressions in the polynomial are called the terms of the polynomial in one variable.

Coefficients

$a_{0},$ $a_{1},$ $a_{2},$ $\cdots$ $a_{n-2},$ $a_{n-1}$ and $a_{n}$ are constants, they are multiplied by the factors in variable form $1,$ $x,$ $x^2,$ $\cdots$ $x^{n-2},$ $x^{n-1}$ and $x^{n}$ respectively. Hence, $a_{0},$ $a_{1},$ $a_{2},$ $\cdots$ $a_{n-2},$ $a_{n-1}$ and $a_{n}$ are called the coefficients of them.

Simple form

It is not convenient to write the polynomial in either ascending or descending order.

$a_{n}\,x^{n}$ $+$ $a_{n-1}\,x^{n-1}$ $+$ $a_{n-2}\,x^{n-2}$ $+$ $\cdots$ $+$ $a_{2}\,x^2$ $+$ $a_{1}\,x$ $+$ $a_{0}$

$a_{0}$ $+$ $a_{1}\,x$ $+$ $a_{2}\,x^2$ $+$ $\cdots$ $+$ $a_{n-2}\,x^{n-2}$ $+$ $a_{n-1}\,x^{n-1}$ $+$ $a_{n}\,x^{n}$

instead of them, it is simply written in summation notation.

$\displaystyle \sum_{\displaystyle k = 0}^{\displaystyle n}{a_{k}\,x^{k}}$

In this case, $k$ is a constant and $k \,=\, 0, 1, 2, 3 \cdots n$