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

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.

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.

$a_{0},$ $a_{1},$ $a_{2},$ $\cdots$ $a_{n-2},$ $a_{n-1}$ 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.

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 $k \,=\, 0, 1, 2, 3 \cdots n$

Latest Math Topics

Nov 11, 2022

Nov 03, 2022

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

Oct 24, 2022

Sep 30, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved