What are Different Types of Polynomials?

Classification of types of polynomials is either based on their degree or number of terms. Knowing different types of polynomials allows us to understand how the polynomial behaves, how many roots/zeros it will have and what method can be used to solve them. This knowledge is highly useful for building a solid foundation in advanced algebra. In this article, we will cover different types of polynomials in detail based on their degree and number of terms.

Table of Contents

• Introduction to Polynomials
• Types of Polynomials
• Types of Polynomials Based on Degree
• Types of Polynomials Based on Terms
• Solved Examples on Types of Polynomials

Introduction to Polynomials

A polynomial is described as an algebraic expression consisting of constants, variables and coefficients joined together through addition or subtraction operation. In standard form a polynomial is represented as: P(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + a_n−2xⁿ⁻² + … + a₁x + a₀

where a_i are the coefficients. In standard form polynomial terms are placed in decreasing order of their degree.

Types of Polynomials

Polynomials are broadly classified into two different types based on their degree and number of terms. Let’s understand each of these in detail below.

Types of Polynomials Based on Degree

In a polynomial the degree of a polynomial is the highest exponent of a variable. Based on the degree of a polynomial, it can be classified into following types:

Types	Functions	Examples
Constant	f(x) = ax0 or a	f(x) = 9
Linear	f(x) = ax1 + b	f(x) = 3x + 5
Quadratic	f(x) = ax2 + bx + c	f(x) = 5x2 + 7x + 3
Cubic	f(x) = ax3 + bx2 + cx + d	f(x) = 9x3 + 9x2 + 7x + 3
Quartic	f(x) = ax4 + bx3 + cx2 + dx + e	f(x) = 9x4 + x3 + 9x2 + 7x + 3
Quintic	f(x) = ax5 + bx4 + cx3 + dx2 + ex + f	f(x) = 4x5 + 9x4 + 4x3 + 6x2 + x + 2

Types of Polynomials Based on Terms

Based on the number of terms a polynomial can be classified as follows:

Types	General Form	Examples
Monomial (One Term)	f(x) = axn ; a ≠ 0	f(x) = 2x2
Binomial (2 Terms)	f(x) = axp + bxq	f(x) = 5x3 + 2x
Trinomial (3 Terms)	f(x) = axp + bx + c	f(x) = 5x3 + 2x + 6

Solved Examples on Types of Polynomials

Example 1: Classify the given polynomials based on degree:

i) 4x + 7   ii) 3x²   iii) 8y⁴ + y³ + y

Solution:

i) 4x + 7 is a linear polynomial.
ii) 3x² is a quadratic monomial.
iii) 8y⁴ + y³ + y is a quartic polynomial.

Example 2: State the number of terms and degree of the following polynomials and tell whether they are monomial, binomial or trinomial:

a) 5x² - 3x + 1   b) 7y   c) x² + 9   d) -3

Solution:
a) 5x² - 3x + 1 has 3 terms and degree = 2. It is a trinomial.
b) 7y has only one term and degree = 1. It is a monomial.
c) x² + 9 has two terms and degree = 2. It is a binomial.
d) -3 is a constant or zero polynomial with one term and degree 0.

Example 3: Classify the given polynomials based on degree and terms together:

• 9z² + 3z + 1
• 7y³ + y
• 5x
• 14x⁴

Solution:
9z² + 3z + 1 is a quadratic trinomial.
7y³ + y is a cubic binomial.
5x is a linear monomial.
14x⁴ is a quartic monomial.

Example 4: Verify whether (x + 1)² is a binomial or trinomial.

Solution:
(x + 1)² = x² + 2x + 1

Therefore, the given polynomial is a trinomial.