# Monomials, Binomials, Trinomials and Polynomials

Before we move to the terms of an algebraic expression, you need to recall the definition of an algebraic expression. As discussed in the previous topic, an algebraic expression is an amalgam of variables and constants of 1 or more terms. The expressions are separated by symbols or operations like (+, –, × and ÷). These expressions are categorized as a monomial, binomial, trinomial and polynomial. Let us know more!

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## Introduction

Before we move any further, let us take help of an example for better understanding. Here are some examples of algebraic expressions.

• 7b + 5m,
• 5x + 3y + 10,
• 5x/y + 3,
• x + y + z,

## Terms of An Algebraic Expression

Every part of an algebraic expression that is separated by a minus or plus sign is known as the term of the algebraic expression. Please note that multiplication and division sign does not separate the terms of an algebraic expression.

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### Types of Algebraic Expressions

Broadly, there are five types of algebraic expressions. These include:

• Monomial
• Binomial
• Trinomial
• Polynomial
• Multinomial

However, here in this topic, we’ll study about the first 4 categories of an algebraic expression.

#### Monomial

A monomial is an algebraic expression that has only one non zero term. Examples of a monomial expression:

• a is a monomial in one variable a
• 10ab2 is a monomial in two variables a and b.
• 5m2n is a monomial in two variables m and n.
• -7pq is a monomial in two variables p and q.
• 5b3c is a monomial in two variables b and c.
• 2b is a monomial in one variable b.
• 2ax/3y is a monomial in three variables a, x and y.
• k2 is a monomial in one variable k.

#### Binomial

A binomial is an algebraic expression that has two non-zero terms. Examples of a binomial expression:

• a2 + 2b is a binomial in two variables a and b.
• 5x3 – 9y2 is a binomial in two variables x and y.
• -11p – q2 is a binomial in two variables p and q.
• m + n is a binomial in two variables m and n.
• b3/2 + c/3 is a binomial in two variables b and c.
• 5m2n2 + 1/7 is a binomial in two variables m and n.

Trinomial

A trinomial is an algebraic expression that has three non-zero terms. Examples of a trinomial expression:

• x + y + z is a trinomial in three variables x, y and z.
• 2a2 + 5a + 7 is a trinomial in one variables a.
• xy + x + 2y2 is a trinomial in two variables x and y.
• -7m5 + n3 – 3m2n2 is a trinomial in two variables m and n.
• 5abc – 7ab + 9ac is a trinomial in three variables a, b and c.
• x2/3 + ay – 6bz is a trinomial in five variables a, b, x, y and z.

#### Polynomial

A polynomial is an algebraic expression that has one, two or more terms. Examples of polynomials:

• 2a + 5b is a polynomial of two terms in two variables a and b.
• 3xy + 5x + 1 is a polynomial of three terms in two variables x and y.
• 3y4 + 2y3 + 7y2 – 9y + 3/5 is a polynomial of five terms in two variables x and y.
• m + 5mn – 7m2n + nm2 + 9 is a polynomial of four terms in two variables m and n.
• 3 + 7x5 + 4x2 is a polynomial of three terms in one variable x.
• 3 + 5x2 – 4x2y + 5xy2 is a polynomial of three terms in two variables x and y.
• x + 5yz – 7z + 11 is a polynomial of four terms in three variables x, y and z.
• 1 + 2p + 3p2 + 4p3 + 5p4 + 6p5 + 7p6 is a polynomial of seven terms in one variable p.

Learn about Different types of Polynomials here.

## Question For You

Question 1: Identify the Monomial, Binomial, Trinomial, Polynomial, and Multinomial from the following:

a) 5pq                      b) 3b + 5c

c) x + y + z             d) a2+ 2b

Answer : a) Monomial             b) Polynomial

c) Trinomial              d) Binomial

Question 2: Explain monomial with example?

Answer: A monomial refers to an expression that involves one term, like 3xy. Monomials include variables, numbers, and whole numbers whose multiplication takes place together. Any number, all by itself, can be a monomial, like the number 5 or the number 2,700.

Question 3: Explain the difference between polynomial and monomial?

Answer: A monomial refers to an expression that comprises of only one term. A polynomial, on the other hand, refers to an expression that has at least one algebraic term, but it does not give any indication of division by a variable or involves variables with fractional exponents.

Question 4: Explain the greatest monomial factor?

Answer: In order to find the greatest common factor (GCF) among monomials, one must take each monomial and write down its prime factorization. Afterwards, one must identify the factors that are common to each monomial. Then one must do multiplication of those common factors together.

Question 5: Is it possible for a degree of a monomial to be negative?

Answer: It is not possible for a degree of a monomial to be a negative value. A monomial is a single term that involves any combination of non-negative powers of variables and constants.

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