August 8, 2024

# Polynomials : Definition, Types of polynomials and Examples, Degree of a polynomial

Polynomials:  An algebraic expression is an expression which is made up of variables and constants along with some algebraic operations.  The several parts of an algebraic expression seperated by + or – operations are called the terms of the  expression. e.g.

(i) $ax^{3}+by^{2}x+cz^{2}$  is  an algebraic expression with three terms  and three variables $x,y,z$

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(ii) $x^{2}+2xy+y^{2}$  is  an algebraic expression with three terms  and two variables $x,y$.

(iii) $2\sqrt{x}+3x$   is  an algebraic expression with two terms  and one variable $x$.

(iv)$\frac{1}{x}$      is  an algebraic expression with one terms  and one variable.

In an algebraic expression , if the powers of variables are non-negative integers , then it is a polynomial.

In the above examples , (i) and (ii) are polynomials, where as (iii) and (iv) are not polynomials.

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# Polynomials in one variable :

Polynomials in one variable are algebraic expressions that consists of  terms in the form of $ax^{n}$, where  $n$ is non-negative integer and a is constant . e.g. $x^{3},&space;\,&space;3x^{2}+2x-2,&space;\,&space;-7x^{6}+10&space;,$ all are polynomials  in variable $x$.

Each term of a polynomial has a  coefficient . so in $-x^{3}+\sqrt{2}x+3$, the  coefficient of $x^{3}$ is -1, coefficient of $x$ is $\sqrt{2}$ and coefficient of $x^{0}$ is 3.

## Degree of a polynomial:

The degree of a polynomial in a single  variable $x$ is the highest power of $x$ in its expression. e.g. $7x^{5}-4x^{2}+5$ is a polyn0mial of degree 5 and $x^{6}-100$ is a polynomial of degree 6.

In general  any polynomial of degree $n$ is an expression of the form

$\large&space;p(x)=&space;a_{0}+a_{1}x+a_{2}x^{2}+............+a_{n-1}x^{n-1}+a_{n}x^{n},$   where   $\large&space;a_{0},\,&space;a_{1},&space;\,&space;a_{2},..........,&space;a_{n-1}&space;,&space;a_{n}$ are constants ,  $a_{n}\neq&space;0$  and $n$ is a non-negative integer .

Here $a_{0}$ is called the constant term of the polynomial and $a_{1},&space;a_{2},a_{},............,a_{n-1},a_{n}$ are called the coefficient of $x,x^{2},&space;x^{3},...........,&space;x^{n-1}&space;\,&space;\,&space;and&space;\,&space;\,&space;x^{n}$ respectively.

In particular if all the constants are zero , then we get $\large&space;p(x)=0$ the zero polynomial.  Zero polynomial has no non-zero terms so the degree of zero polynomial is not defined.

### Constant polynomial:

A polynomial containing only the constant term is called constant polynomial. e.g. $\sqrt{2},&space;3,-7$ all are constant polynomials.

Let $p(x)=a\,&space;,\,&space;a\neq&space;0$  is a non-zero constant polynomial ,

then $p(x)$ can be written as  $p(x)=ax^{0}$

$\Rightarrow&space;deg\,&space;\,&space;p(x)=0$

Therefore the degree of any non-zero constant polynomial is zero.

Based  on the number of terms,  polynomials are classified as

(i) A polynomial containing one term  is called a monomial. e.g. $5x^{3},&space;\,&space;-9x,&space;\,&space;7$ etc. all are monomials.

(ii) A polynomial containing two terms  is called a binomial. e.g. $9x^{2}+2,\,&space;\,&space;-7x^{5}+3x$ etc.

(iii)A polynomial containing three terms  is called a trinomial. e.g.  $\sqrt{5}x^{3}+7x-2,&space;\,&space;\,&space;2x^{3}+7x+9,&space;\,&space;x^{25}-10x+5$ etc. all are trinomials.

#### Linear polynomial :

A polynomial of degree one is called  a linear polynomial. A linear polynomial in $x$ is  of the form

$\large&space;p(x)=ax+b,&space;\:&space;\,&space;where&space;\,&space;\,&space;a,&space;\,&space;b&space;\,&space;\,&space;are&space;\,&space;\,&space;constants&space;\,&space;\,&space;and&space;\,\,&space;a&space;\neq&space;0$.

Any linear polynomials in $x$ have  at most two terms . e.g. $7x+2,&space;\,&space;-2x,\,&space;\sqrt{2}x-6$ all are linear polynomials.

A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial in one variable will have at most tree terms.  Any quadratic polynomial in $x$ will be of the form    $\large&space;ax^{2}+bx+c\,&space;\,&space;where&space;\,&space;\,&space;a\neq&space;0\,&space;\,&space;and&space;\,&space;\,&space;a,b,c\,&space;\,&space;are\,&space;\,&space;constants.$

e.g. $4x+&space;5x^{2},&space;x^{2},&space;x^{2}-x+6$ all are quadratic polynomials.

###### Cubic polynomial :

A polynomial of  degree  3 is called  cubic polynomials. Any  cubic  polynomial can have at  most 4 terms.  $6x^{3}-6x,&space;2x^{3}+3x^{2}-5x+2,&space;4x^{3}&space;,&space;x^{3}-2x^{2}$ all are examples of cubic polynomials.

#### Bina singh

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## One thought on “Polynomials : Definition, Types of polynomials and Examples, Degree of a polynomial”

1. Divyanshi says:

Very good explanation & examples

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