Division of Algebraic Expressions
Division of algebraic expressions is performed in the same way as division is performed on two whole numbers or fractions. Division of two algebraic expressions or variable expressions involves taking out of common terms and canceling them out. These common terms either include constants, variables, terms, or just coefficients. In this article, let us learn about the rules of dividing different types of algebraic expressions with solved examples.
What is Division of Algebraic Expressions?
In the division of an algebraic expression, we cancel the common terms, which is similar to the division of the numbers. Division of algebraic expressions involves the following steps.
 Step 1: Directly take out common terms or factories the given expressions to check for the common terms.
 Step 2: Cancel the common term.
Note: Here, the common terms correspond to either of the following: constants, variables, terms, or just coefficients.
There are different types of division of algebraic expressions.
 Division of monomial by a monomial
 Division of polynomial by a monomial
 Division of polynomial by a polynomial
In any case, we first take out common terms from the given polynomials and then eliminate that common term/terms. Let us discuss them case by case.
Division of Monomial by a Monomial
A monomial is a type of expression that has only one term. The correct method to perform the division of monomial by another monomial is given below:
Consider an example, 27x^{3}÷3x
Here 3x and 27x^{3} be the two monomials.
 Write their prime factorization. 27x^{3 }÷ 3x = 27×x×x×x/3×x
 Cancel the common term, which is 3x.
Thus, 27x^{3 }÷ 3x = 9x^{2}
Division of Polynomial by a Monomial
A polynomial contains a few types of expressions, some of which are binomial, trinomial, or an equation with nterms.
Now, let's perform the dividing polynomials by monomials.
(4y^{3 }+ 5y^{2 }+ 6y) ÷ 2y
Here, the trinomial is 4y^{3 }+ 5y^{2 }+ 6y, and the monomial is 2y.
 In trinomial, on taking the common factor 2y, it becomes: 4y^{3 }+ 5y^{2 }+ 6y = 2y(2y^{2 }+ (5/2)y + 3)
 Now, we do the division operation: {2y(2y^{2 }+ (5/2)y + 3)} ÷ 2y. Cancel 2y from the numerator and the denominator: (4y^{3 }+ 5y^{2 }+ 6y) ÷ 2y = 2y^{2 }+ (5/2)y + 3
Thus, (4y^{3 }+ 5y^{2 }+ 6y) ÷ 2y = 2y^{2 }+ (5/2)y + 3
Division of Polynomial by a Polynomial
Let us consider polynomials that divide polynomial for performing the division operation.
(7x^{2 }+ 14x) ÷ (x + 2)
Here, both polynomials exist in the binomial form.
 Take out the common factors. For the polynomial 7x^{2 }+ 14x, x is the common factor.
 So, consider “7x” as a common factor among them. Then it becomes, 7x^{2 }+ 14x = 7x(x+2)
 Now, (7x^{2 }+ 14x) ÷ (x + 2) = 7x(x + 2) / (x + 2)
 Eliminate (x+2) from the numerator and denominator, we get the solution for the long dividing polynomials as: (7x^{2 }+ 14x) ÷ (x + 2) = 7x
Thus, (7x^{2 }+ 14x) ÷ (x + 2) = 7x
Topics Related to Division of Algebraic Expressions
 Factorization of algebraic expressions
 Subtraction of Algebraic Expressions
 Addition of Algebraic Expressions
 Multiplication of Algebraic Expressions
Important Notes
 We can only add and subtract like or similar terms.
 We can multiply both like and unlike terms.
 Division of algebraic expression is done by using algebraic identities.
 Remember the negative sign also distributes to all terms in the bracket.
Examples on Division of Algebraic Expressions

Example 1: Rose wants to divide the polynomial (4x^{3}−3x^{2}+4x) by the monomial 2x. Can you help her with the solution?
Solution:
Here, the polynomial is 4x^{3}−3x^{2}+4x, and the monomial is 2x.
On taking the common factor 2x from the numerator, it becomes:4x^{3}−3x^{2}+4x=2x(2x^{2}−(3/2)x+2)
Now, we divide the expression by 2x:
(4x^{3}−3x^{2}+4x)÷2x={2x(2x^{2}−(3/2)x+2)}÷2xOn canceling the 2x from the numerator and the denominator, we get:
2x^{2}−(3/2)x+2
Therefore, (4x^{3}−3x^{2}+4x)÷2x=2x^{2}−(3/2)x+2

Example 2: Can we help Ashley to find the solution of (24a^{2}+48a)÷(6a+12) by using the division of a polynomial by a polynomial method?
Solution:
We have to find: (24a^{2}+48a)÷(6a+12)
We will first find common factors for the polynomials (24a^{2}+48a).
Here, 4a is the common factor. Hence, it becomes,24a^{2}+48a=4a(6a+12)Let us do the division of algebraic operation,
(24a^{2}+48a)÷(6a+12)=[4a(6a+12)]÷(6a+12)
On eliminating (6a+12) from the numerator and denominator, we get the solution for the division.
Thus, [4a(6a+12)]÷(6a+12)=4aTherefore, (24a^{2}+48a)÷(6a+12)=4a
FAQs on Division of Algebraic Expressions
What Is Division of Algebraic Expressions in Math?
Division of an algebraic expression is similar to the division of numbers. In this method, we carry out division by taking out common terms and get a simplified answer to division of given algebraic expressions,
What Are the Steps in Division of Algebraic Expressions?
The steps to divide algebraic expressions are:
 Step 1: Directly take out common terms or factorize the given expressions to check for the common terms.
 Step 2: Cancel the common term.
How To Divide Two Algebraic Expressions?
In the division of two algebraic expressions, we cancel the common terms from the expressions, which is similar to the division of the numbers.
What Are the Steps To Division of Rational Algebraic Expressions?
Few steps to divide rational algebraic expressions are:
 Step 1: Check out the factors of both the numerators and denominators of all the given fractions.
 Step 2: Change the division sign into a multiplication sign, reciprocate the fraction and further multiply the terms.
 Step 3: Reduce the given fractions.
What Is the Result of Division of Polynomial by a Polynomial?
The result of the division of polynomial by a polynomial gives the answer to the division of two polynomials, which is performed using the steps given below:
 Check for the given polynomials.
 Take out the common factors.
 Eliminate that common factor from the numerator and denominator, we get the solution for the long division of two given polynomials.
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