June 14, 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.

(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.

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}, \, 3x^{2}+2x-2, \, -7x^{6}+10 , 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 p(x)= a_{0}+a_{1}x+a_{2}x^{2}+............+a_{n-1}x^{n-1}+a_{n}x^{n},   where   \large a_{0},\, a_{1}, \, a_{2},.........., a_{n-1} , a_{n} are constants ,  a_{n}\neq 0  and n is a non-negative integer .

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

In particular if all the constants are zero , then we get \large 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}, 3,-7 all are constant polynomials.

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

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

\Rightarrow deg\, \, 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}, \, -9x, \, 7 etc. all are monomials.

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

(iii)A polynomial containing three terms  is called a trinomial. e.g.  \sqrt{5}x^{3}+7x-2, \, \, 2x^{3}+7x+9, \, 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 p(x)=ax+b, \: \, where \, \, a, \, b \, \, are \, \, constants \, \, and \,\, a \neq 0.

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

Quadratic polynomial : 

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 ax^{2}+bx+c\, \, where \, \, a\neq 0\, \, and \, \, a,b,c\, \, are\, \, constants.

e.g. 4x+ 5x^{2}, x^{2}, 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, 2x^{3}+3x^{2}-5x+2, 4x^{3} , x^{3}-2x^{2} all are examples of cubic polynomials.

Different types of polynomials


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