9th Maths

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.