Polynomial: 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) is an algebraic expression with three terms and three variables .
(ii) is an algebraic expression with three terms and two variables .
(iii) is an algebraic expression with two terms and one variable .
(iv) 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 , where is non-negative integer and a is constant . e.g. all are polynomials in variable .
Each term of a polynomial has a coefficient . so in , the coefficient of is -1, coefficient of is and coefficient of is 3.
Degree of a polynomial:
The degree of a polynomial in a single variable is the highest power of in its expression. e.g. is a polyn0mial of degree 5 and is a polynomial of degree 6.
In general any polynomial of degree is an expression of the form
where are constants , and is a non-negative integer .
Here is called the constant term of the polynomial and are called the coefficient of respectively.
In particular if all the constants are zero , then we get , the zero polynomial. Zero polynomial has no non-zero terms so the degree of zero polynomial is not defined.
A polynomial containing only the constant term is called constant polynomial. e.g. all are constant polynomials.
Let is a non-zero constant polynomial ,
then can be written as
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. etc. all are monomials.
(ii) A polynomial containing two terms is called a binomial. e.g. etc.
(iii)A polynomial containing three terms is called a trinomial. e.g. etc. all are trinomials.
Linear polynomial :
A polynomial of degree one is called a linear polynomial. A linear polynomial in is of the form
Any linear polynomials in have at most two terms . e.g. 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 will be of the form
e.g. all are quadratic polynomials.
Cubic polynomial :
A polynomial of degree 3 is called cubic polynomials. Any cubic polynomial can have at most 4 terms. all are examples of cubic polynomials.