Division of Algebraic Expression

Numbers are the basic units to accurately express a quantity in mathematical terms. Along with this even alphabets are used in cases of problem-solving, decision-making, and word problems. And, variables are symbols used in denoting numbers, alphabets and other arbitrary elements. Combining all, when a statement or a mathematical condition has numbers, alphabets, variables, or any other special character, then this is termed to be as an Algebraic expression. Addition, subtraction, multiplication, and division are the 4 operations that function as an algebraic expression. 

From here on, let’s learn the important pointers about the division of algebraic expression only with examples. 

A Brief Idea about Division of Algebraic Expression

Any algebraic expression will consist of numbers, operations, variables and a few special characters used in the field of mathematics. Since the sequence is formed using constants and variables, this formation is referred to as a mathematical expression. 

Note that no algebraic expression is found to have equal signs or sides. Examples for algebraic expressions are 4x + 5y – 3, 2x – 5, 4x2 – 1xy + 6, 6x3 – 2x2 + 6x – 1 and so on. 

Lastly, the division of algebraic expression is the inverse proportion of multiplication. In the classification of algebraic expressions, there are 3 majors namely Monomial, Binomial and Polynomial. 

We are going to understand each concept in short as described in the following sections.

Defining what is a Monomial and a Polynomial

Polynomial is the term used to denote the condition of P = 0. Here, this formula is also called the Polynomial equation or Polynomial expression. A polynomial can range anywhere between a trinomial, binomial or even possess ‘n’ count of terms. 

When a polynomial formula has only 1 term, then this is said to be the monomial. 15xy is an example of a monomial term.

Binomial, as the name says, is a polynomial equation that has only 2 terms. To state an example, 1x + 4. Take the case of 2a(a+b) 2. This is also considered as a binomial expression due to the presence of binomial factors ‘a’ and ‘b’.

The Process of Dividing a Monomial by Another Monomial

Consider this division of polynomial expression 12a3 ÷ 4a. Here, the terms 4a and 12a3 are the 2 monomials of the equation. One can quickly simplify the equation by cancelling common values. This is quite similar to that of natural division in real numbers.

12a3 ÷ 4a = (12 × a × a × a) / 4 × a

After the cancellation process, we will get 12a3 ÷ 4a = 3a2.

When a Monomial Divides a Polynomial

As we read before, a polynomial equation can have multiple options such as 2, 3 or ‘n’. For this condition, we are considering a trinomial. We will take a monomial “3a” and then divide it using the polynomial equation “(6a3 + 7a2 + 9a)”.

(6a3 + 7a2 + 9a) ÷ 3a

Take only the common factors from the equation and for our case, 3a is the common value. So the equation will note become:

(6a3 + 7a2 + 9a) = 3a (2a2 + 7/3a + 3)

Start dividing the complete set by the monomial 3a. 

(6a3 + 7a2 + 9a) ÷ 3a = 3a (2a2 + 7/3a + 3) / 3a

Upon cancellation of the terms 3a and 3a from the denominator and the numerator, we will get:

(6a3 + 7a2 + 9a) ÷ 3a = (2a2 + 7/3a + 3)

Hence, we divided a polynomial expression using a monomial term. Finally, we are about to learn the division of algebraic expression between 2 different polynomial values as given below.

Division Operation Between 2 Polynomials

We will consider this case using simple numbers for a better understanding. Take 2 separate polynomial figures and divide them.

(6a2 + 12a) ÷ (a + 2) 

Did you notice that the above 2 polynomials are in the monomial form? Now, pick the common factors, which is 6a, and the equation becomes:

(6a2 + 12a)  = 6a (a + 2)

As you might have guessed, divide the complete set with the monomial (a + 2).

(6a2 + 12a) ÷ (a + 5) = 6a (a + 2) / (a + 2)

By cancelling the common terms (a +2), we get (6a2 + 12a) ÷ (a + 5) = 6a

Thus, we formed a proper equation, resulting in a monomial value.


An Algebraic Expression is defined as a mathematical sequence, formulated using operational symbols, numbers, constants and variables. The Division of Algebraic Expression refers only to diving between the various types of expressions. Monomials, Binomials, and trinomials are classified as Polynomials for an algebraic equation. Monomial has only 1 term and Binomial has 2 terms, while a polynomial can have ‘n’ count of values. The division can happen between 2 monomials, 2 binomials, 2 polynomials or even crossing 1 type from each other. Always, start the division process by identifying the common terms and cancelling them from the numerator and denominator. 

FAQ (Frequently Asked Questions)

1. Can 2 Binomial Terms have the Same Value?

Yes. 2 binomial terms can have the same value, concerning the numbers and not the term itself. For example, 7xy + 3. 

2. What is a Numeric Expression?

When an expression consists of only numerical digits, then it is said to be a numeric expression. 10 - 4, 3 + 6, etc. are the common examples. Numeric expressions may have different signs and any count of root terms. Again, the operations can also vary how many ever times as stated.

3. Define what is Meant by a Variable Expression.

When a mathematical sequence has only variables and alphabets, then it is a variable expression. Examples are 2x - 4, 3axy + 7, 4x/3y and such. A variable expression can have any operational question, any sign and multiple root values.

4. 8xy3 + 9x2y3 - Is this a Trinomial Algebraic Expression?

No. 8xy3 + 9x2y3 is a binomial algebraic expression since there are only 2 different entities in the equation, that is “8xy3” and “9x2y3” respectively.

5. How can I Find the Quotient of 2 Different Monomials?

Note that the quotient of 2 distinct monomials is also a monomial. So, you can find the quotient of these 2 different monomials by multiplying the quotient of numerical coefficients with its variables.

Quotient of two monomials = (the quotient of their numerical coefficients) x (the quotient of their variables).

6. What are the 4 Rules for any Algebraic Expression?

The 4 major rules for any algebraic expression are Commutative Property of Addition, Associative Property of Addition, Associative Property of Multiplication, and Commutative Property of Multiplication. There is also one more additional norm called the Distributive Properties of Addition Over Multiplication.

7. Can the Division of an Algebraic Expression Present without an Operational Symbol?

No. The division of algebraic expressions should have at least 1 operational symbol and 1 variable with a minimum significance like this “3(x + 8y)”.