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) 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.
Also Read :
- Factoring Polynomials Formula and Factoring Polynomials Worksheet with Answers
- Factoring trinomials worksheet