Introduction of Polynomials
Polynomials = Poly (means many) + nomials (means terms). Thus, a polynomial contains many terms
Thus, a type of algebraic expression with many terms having variables and coefficients is called polynomial.
Let us consider another example, in this ‘x’ is called variable.
Power of ‘x’, i.e. 2 is called exponent.
Multiple of ‘x’, i.e. 2 is called coefficient.
The term ‘2’ is called constant.
And all items are called terms.
Let us consider another example –
In this there are two variables, i.e. x and y. Such polynomials with two variables are called Polynomials of two variables
Power of x is 2. This means exponent of x is 2.
Power of y is 1. This means exponent of y is 1.
The term ‘5’ is constant.
There are three terms in this polynomial.
Types of Polynomial:
Monomial – Algebraic expression with only one term is called monomial.
Binomial: Algebraic expression with two terms is called binomial.
Trinomial – Algebraic expression with three terms is called trinomial.
But algebraic expressions having more than two terms are collectively known as polynomials.
Variables and polynomial:
Polynomial of zero variable
If a polynomial has no variable, it is called polynomial of zero variable. For example – 5. This polynomial has only one term, which is constant.
Polynomial of one variable
Polynomial with only one variable is called Polynomial of one variable.
In the given example polynomials have only one variable i.e. x, and hence it is a polynomial of one variable.
Polynomial of two variables
Polynomial with two variables is known as Polynomial of two variables.
In the given examples polynomials have two variables, i.e. x and y, and hence are called polynomial of two variables.
Polynomial of three variables
Polynomial with three variables is known as Polynomial of three variables.
In the above examples polynomials have three variables, and thus are called Polynomials of three variables.
In similar way a polynomial can have of four, five, six, …. etc. variables and thus are named as per the number of variables.
Degree of Polynomials:
Highest exponent of a polynomial decides its degree.
Polynomial of 1 degree:
Example: 2x + 1
In this since, variable x has power 1, i.e. x has coefficient equal to 1 and hence is called polynomial of one degree.
Polynomial of 2 degree
In this expression, exponent of x in the first term is 2, and exponent of x in second term is 1, and thus, this is a polynomial of two(2) degree.
To decide the degree of a polynomial having same variable, the highest exponent of variable is taken into consideration.
Similarly, if variable of a polynomial has exponent equal to 3 or 4, that is called polynomial of 3 degree or polynomial of 4 degree respectively.
Important points about Polynomials:
- A polynomial can have many terms but not infinite terms.
- Exponent of a variable of a polynomial cannot be negative. This means, a variable with power - 2, -3, -4, etc. is not allowed. If power of a variable in an algebraic expression is negative, then that cannot be considered a polynomial.
- The exponent of a variable of a polynomial must be a whole number.
- Exponent of a variable of a polynomial cannot be fraction. This means, a variable with power 1/2, 3/2, etc. is not allowed. If power of a variable in an algebraic expression is in fraction, then that cannot be considered a polynomial.
- Polynomial with only constant term is called constant polynomial.
- The degree of a non-zero constant polynomial is zero.
- Degree of a zero polynomial is not defined.