Polynomials
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.
Example: `3x`, `5y^2+2x+5`, and `2x^2+2`
Let us consider another example, `2x^2+2` 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: `5x^2+2x+5`
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.
Example: `2x, 2, 5x, 3y`, etc.
Binomial
Algebraic expression with two terms is called binomial.
Example: `2x+2, 3y^2+5, 3m+3`, etc.
Trinomial
Algebraic expression with three terms is called trinomial.
Example: `3x+3y+2, 5y^2+2y+2`, etc.
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.
Example: `5x+2, 2x^2+x+3`, etc.
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.
Example: `5x+y, 2x+3y+2`, etc.
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.
Example: `2x^2+3y+m+2` and `5y+3m+z+5`
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
Example: `2x^2+2x+2`
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.