Types of Polynomials

In algebra, a polynomial is an expression that is made up of variables and constants in which the exponents of the variables are only positive integers and not fractions. The polynomial terms are separated mostly by either addition or subtraction operators. Polynomials find their applications in writing polynomial equations and defining polynomial functions. An example of a polynomial expression is x²+2x-3. This polynomial has a degree of 2 since the term with the highest power of 'x' is 'x²'. This polynomial has 3 terms. Polynomials can be categorized with respect to their total number of terms and their degree. 

1.	Polynomials Definition
2.	Types of Polynomials Based on Degree
3.	Types of Polynomials Based on Terms
4.	Solved Examples on Types of Polynomials
5.	Practice Questions on Types of Polynomials
6.	FAQs on Types of Polynomials

Polynomials are algebraic expressions in which the variables have only non-negative integer powers. For example,: 5x² - x + 1 is a polynomial. The algebraic expression 3x³ + 4x + 5/x + 6x^3/2 is not a polynomial, since one of the powers of 'x' is a fraction and the other is negative. Polynomials are expressions with one or more terms having a non-zero coefficient. The terms comprise variables, exponents, and constants. The first term of the polynomial is called the leading term. A standard polynomial is one where the highest degree is the first term, and the subsequent terms are arranged in descending order of the powers or the exponents of the variables followed by constant values. The number multiplied with a variable is called the coefficient.  The number without any variable is called a constant.

The power of the leading term or the highest power of the variable is called the degree of the polynomial. This is obtained by arranging the polynomial terms in the descending order of their powers. Based on the degree of the polynomial, they can be classified into 4 major types. They are,

• Zero or Constant polynomial
• Linear polynomial
• Quadratic polynomial
• Cubic polynomial

Look at the table given below to understand the meaning of the types of polynomials with examples.

Type of Polynomial	Meaning	Examples
Zero or constant polynomial	Polynomials with 0 degrees are called zero polynomials.	3 or 3x0
Linear polynomial	Polynomials with 1 as the degree of the polynomial are called linear polynomials. In linear polynomials, the highest exponent of the variable(s) is 1	x + y - 4, 5m + 7n, 2p
Quadratic polynomial	Polynomials with 2 as the degree of the polynomial are called quadratic polynomials.	8x2 + 7y - 9, m2 + mn - 6
Cubic polynomial	Polynomials with 3 as the degree of the polynomial are called cubic polynomials.	3x3, p3 + pq + 7

There are different types of polynomials with respect to their number of terms. There are polynomials with one term, two terms, three terms, and even more. Based on the number of terms, polynomials are classified as:

• Monomials: A monomial is a polynomial expression that contains only one term. For example 4t, 21x, 2y, 9pq.  Furthermore, 2x + 5x + 10x is a monomial because these are like terms added together that result in 17x.
• Binomials: A binomial is a polynomial with two, unlike terms. For example, 3x + 4x² is a binomial as it contains two unlike terms, that is, 3x and 4x². and 10pq + 13p².
• Trinomials: A trinomial is a polynomial with three, unlike terms. For example, 3x + 5x² – 6x³ and 12pq + 4x² – 10.

We can also have more than 3 terms in a polynomial expression. Polynomials that have 4 unlike terms are called four-term polynomials. Similarly, polynomials with 5 terms are called five-term polynomials, and so on.
Important Notes

• In a polynomial, two terms are separated by the addition or subtraction sign. Multiplication and division operators are not used to create more terms in a polynomial. For example, 3xy is considered as 3 × x × y, is a monomial, whereas 3x+y is a binomial.
• Based on the terms in a polynomial, it can be classified into the following 3 types. They are monomial, binomial, trinomial.
• Based on the degree of a polynomial, it can be classified into 4 types. They are zero polynomial, linear polynomial, quadratic polynomial, cubic polynomial.
• Polynomials should have a whole number as the degree. Expressions with negative exponents are not polynomials. For example, x^-2 is not a polynomial.
• Polynomials do not have variables in their denominator. For example, \(\dfrac{2}{x+2}\) is not a polynomial.

Topics Related to Types of Polynomials

Check out some interesting articles related to types of polynomials. 

• Algebra
• Polynomial Calculator
• Polynomial Functions
• Polynomial Expressions
• Value of a Polynomial
• Polynomial Identity
• n^th Degree Polynomial

 

What are the Types of Polynomials?

A polynomial is an expression that is made up of variables and constants. Polynomials are categorized based on their degree and the number of terms. Based on the number of terms in a polynomial, there are 3 types of polynomials. They are monomial, binomial and trinomial. Based on the degree of a polynomial, they can be categorized as zero or constant polynomials, linear polynomials, quadratic polynomials, and cubic polynomials. 

What are the Types of Polynomials Based on the Number of Terms?

Polynomials are categorized according to the number of terms as follows. They are,

• Monomials: A monomial is a polynomial expression that contains only one term. For example, 2x + 5x + 10x is a monomial because when we add the like terms it results in 17x. Furthermore, 4t, 21x, 2y, 9pq are also monomials.
• Binomials: A binomial is a polynomial with two, unlike terms. For example, 3x + 4x² is a binomial as it contains two unlike terms, that is, 3x and 4x². Likewise, 10pq + 13p² is also a binomial.
• Trinomials: A trinomial is a polynomial with three, unlike terms. For example, 3x + 5x² – 6x³ is a trinomial. Likewise, 12pq + 4x² – 10 is also a trinomial.

In a polynomial, there should not be any negative number as the exponent of the variable. The variable should not be in the denominator of the polynomial that is in fractional form.

What are the Types of Polynomials Based on Degree?

Polynomials can be classified on the basis of their degree as follows. They are, 

• Zero or Constant polynomial: Polynomials with 0 degrees are called zero polynomials. For example, 2 or 2y⁰.
• Linear polynomial: Polynomials with 1 as the degree of the polynomial are called linear polynomials. In linear polynomials, the highest exponent of the variable(s) is 1. For example, p + q - 2.
• Quadratic polynomial: Polynomials with 2 as the degree of the polynomial are called quadratic polynomials. For example, 8x² + 7xy + 3.
• Cubic polynomial: Polynomials with 3 as the degree of the polynomial are called cubic polynomials. For example. 3x³ - 1.

What is a Polynomial with 5 Terms Called?

Polynomials with 5 terms are known as five-term polynomials. For example, 3x³-xy+y²+6x²-8.

What is a Polynomial with 4 Terms Called?

Polynomials with 4 terms are known as quadrinomials. For example, x³+3x-2y+5.