Polynomial: Definition and Types

 

Polynomials

The product form of any number or variables is called an algebraic term.

3x, , - 4x²y³ etc. are a few examples of algebraic terms.

Any algebraic term contains three parts – coefficient, base and power. For example, in 7x³, 7 is the coefficient, x is the base and 3 is the power of the base. The sum or difference of two or more algebraic terms form an algebraic expression. For example, 2x + 5 is a binomial expression, x² – 6xy + 9y² is a trinomial expression, etc. However, an algebraic term itself is a monomial expression. x, 2x³, -y² etc. are monomial expression.

The algebraic expressions in which power of variables are whole numbers are known as polynomials.

For an expression to be a polynomial, its variable –

1. Must have a whole number power. Eg. x², 2x³ + 4, 3y⁵ – 2y³ etc.
2. Must not have a power as negative integers or decimal value. Eg. , ,  etc.

3.       The variable must not be in denominator or root form which can’t be solved. Eg.  ,  + 4,  etc.

Some Examples of ‘polynomials’ or ‘non polynomials.’

a.  4x + 7                                              polynomial                              

b.  5x² – 2x + 3                                     polynomial      

c.  y +                                                 non-polynomial

d.   x + 3                                          polynomial      

e.   – 2x + 1                                        polynomial      

f.   – 7                                        non-polynomial

Types of polynomials

1.  Monomial expression:

An algebraic expression with only one term is called monomial expression. For eg. x, 2x³, -y² etc. are monomial expression.

2.  Binomial Expression:

An algebraic expression with two unlike terms separated by the sign of addition or subtraction is called binomial expression. for eg. 2x + 5, 4x² – 3y, 2xy + 3y etc. are binomial expressions.

3.  Trinomial Expression:

An algebraic expression with three unlike terms separated by the sign of addition or subtraction is called binomial expression. for eg. 2x + 5y – 4, 4a² – 3b³ + 2c, 2xy + 3yz + 4xz etc. are trinomial expressions.

If there are four or more unlike terms in an expression separated by the sign of addition or subtraction then, they are combinedly termed as polynomials. For eg. 2x + 3y – 4z + 5, 4x⁴ + 3x3 + 2x² – x + 5 etc. are some polynomials.

Degree of Polynomials

Let’s consider an algebraic term 7x².

In 7x², variable x is multiplied 2 times. So, the degree of 7x² is 2.

Similarly, in 4y³, the variable y is multiplied 3 times. So, the degree of 4y³ is 3.

In – 6a⁵, the variable a is multiplied 5 times. So, the degree of  – 6a⁵ is 5.

In the case of a term that contains two or more variables, the sum of the powers of each variable is the degree of the term. For examples,

In 2x³y², the sum of powers of x and y is (3 + 2) = 5. So, the degree of 2x³y² is 5.

In 5xyz, the sum of the powers of x, y and z is (1 + 1 + 1) = 3. So, the degree of 5xyz is 3.

In the case of a polynomial, the highest power of its any term is the degree of the polynomial. For example,

In 3x² + 2x – 1

Power of 1^st term = 2

Power of 2^nd term = 1

Power of 3^rd term = 0

Therefore, the degree of a polynomial 3x² + 2x – 1 is 2.

In 2y⁵ – 4y³ + 7y,

Power of 1^st term = 5

Power of 2^nd term = 3

Power of 3^rd term = 1

Therefore, degree of the polynomial 2y⁵ – 4y³ + 7y is 5.

In 3x²y² + 2x³y⁴ – 5xy

Sum of powers of 1^st term = (2 + 2) = 4

Sum of powers of 2^nd term = (3 + 4) = 7

Sum of powers of 3^rd term = (1 + 1) = 2

Therefore, degree of the polynomial 3x²y² + 2x³y⁴ – 5xy is 7.

Name of Polynomials based on the degree

Based on the degree of polynomial, they are given certain names.

Polynomial	Degree	Example
Constant or Zero Polynomial	0	6
Linear Polynomial	1	3x+1
Quadratic Polynomial	2	4x2 + x + 1
Cubic Polynomial	3	6x3 + 4x2 + x + 1
Quartic Polynomial	4	8x4 + 6x3 + 4x2 + x + 1