Division of Algebraic Expressions (original) (raw)
Last Updated : 23 Jul, 2025
Division of algebraic expressions is a key operation in algebra. It is essential for simplifying expressions and solving equations. It is used to perform polynomial long division or synthetic division. Division of algebraic expressions is performed as as division on two whole numbers or fractions. In Math, an algebraic expression is defined as an expression which is made up of constants, variables and arithmetic operators.
In this article, we will learn in detail about the division of algebraic expressions, and their types, and solve problems based on it.
**What is Division of Algebraic Expressions?
The division of algebraic expressions refers to dividing one algebraic expression by another. Algebraic expressions are made up of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Dividing algebraic expressions is comparable to dividing numerical expressions, but it requires additional considerations due to the inclusion of variables. There are different types of algebraic expression which are discussed below:
- **Example of monomial: 4xy, 89z, 56pqr, -9
- **Example of binomial: 34x + 45y, 9xz + 5xy
- **Example of polynomial: w+ x +y +z, 3r+5t-8z-7p
Types of Algebraic Division
Let us consider the following cases for division:
- Division of Monomial by a Monomial
- Division of Polynomial by a Monomial
- Division of Polynomial by a Polynomial
**Division of Monomial by a Monomial
This is the simplest case where both the dividend and divisor are monomials. These are factorized and the common factors are cancelled out.
**Example 1: Divide 24y 3 by 8y
**Solution:
24y3 can be written as 2 × 2 × 2 × 3 × y × y × y
8y can be written as 2 × 2 × 2 × y
So, we need to evaluate (2 × 2 × 2 × 3 × y × y × y)/(2 × 2 × 2 ×y)
Cancelling the common factors,
We obtain the result as 3y2
**Example 2. Divide 20mnz by 4 m
**Solution:
Dividend = 4 × 5 × m × n × z
Divisor = 4 × m
Dividing we get, (4 × 5 × m × n × z)/(4 × m)= 5 n z
**Division of Polynomial by a Monomial
**Example 1: Divide 32(m 2 np+ mn 2 p+mnp 2 ) by 4mnp
**Solution:
Dividend= 32(m2np+ mn2p+mnp2) = 4 × 8 × m n p (m +n+ p)
Divisor= 4mnp
On dividing, (4 × 8 × m n p (m +n+ p)/(4mnp)
= 8(m+ n +p)
**Example 2: Divide 36pqr + 12r by 4r
**Solution:
Dividend =36pqr+ 12r =12 r(3pq+1)
Divisor= 4r
So, (36pqr+ 12r ) / 4r can be written as
= 12 r (3pq+1) /4r
= 4r × 3 × (3pq+1)/4r
Canceling the common terms,
We get, 3(3pq+1)
**Division of Polynomial by a Polynomial
Sometimes there are some common factors between the dividend and divisor. The strategy here also is to cancel the common factors to obtain the quotient.
**Example 1: (12m 2 + 24m)/(m+2)
**Solution:
12m2+24 m can be written as 12m(m+2)
So, (12m2 + 24 m)/(m+2) =12m(m+2)/(m+2)= 12m
So the quotient is 12m
_Check: 12m × (m+2)= 12m 2 +24m
We may apply the long division method also to compute the quotient and remainder. The terms of dividend are divided repeatedly by the first term of the divisor.
**Example 2: Divide m 4 -m 3 +m 2 +5 by m+1
**Solution:
This dividend can be rewritten as
m3(m+1)-2m2(m+1)+3m(m+1)-3(m+1)+8
= m+1(m3-2m2+3m-3)+8
Division by m+1 gives
Quotient =(m3-2m2+3m-3)
Remainder=8
**Example 3: Divide m 4 **- n 4 by m 2 - n 2
**Solution:
By using the identity a2 -b2 = (a-b) (a + b)
We can write m4 - n4 as (m2 - n2)(m2 + n2)
So, cancelling the common terms in dividend and divisor, we get the quotient as m2+n2
Practice Questions on Division of Algebraic Expressions
**Q1. Evaluate: (18y 3 ) / (2y 2 ) when y = 2
**Q2. Evaluate: (40p 4 q 2 ) / (5p^2q) when p = 2 and q = 3
**Q3. Divide the following polynomials and find the quotient and remainder: (10x 2 **- 7x + 8) by (5x - 35x - 3)
**Q4. Determine whether the first polynomial is a factor of the second: (x + 1) in (2x 2 + 5x + 4)
**Q5. Divide the following polynomials and find the quotient and remainder: (3x 2 + 4x + 5) by (x - 2)
**Q6. Divide the following polynomials and find the quotient and remainder: (x 4 - x 3 + 5x) by (x - 1)
**Q7. Is (x + 1) a factor of (2x 2 + 5x + 4)?
**Q8. Is (2a - 3) a factor of (10a 2 - 9a - 5)?
**Q9. Write the degree of 2x + x 2 – 8
**Q10. Evaluate: (72a 5 b 3 c) / (12a 3 b 2 ) when a = 2, b = 1, and c = 3
**Points to Remember
Generally an error is a mistake or difference occurred in the calculated value because of the inaccuracy of the calculated number from its true value.
- Coefficient 1 is usually not written. While adding like terms, do not forget to include it in sum.
**E.g. Sum 4x and x + 4 is (4x + x + 4) i.e. 5x + 4 not 4x + 4.
- While substituting negative values, make use of brackets.
**E.g. Product of -5 and (2x + 3) i.e. -5(2x + 3) is equal to -10x -15 not -10x + 15.
- When an expression written in a bracket is multiplied by a constant outside the bracket, open the brackets carefully. Make sure that each term of expression has to be multiplied by the constant.
**E.g. If it is given that x=-2 then 5x-2 can be written as, 5(-2)-2 which is equal to -12
- When squaring any monomial, both the contact coefficient and the variable are squared.
**E.g. Square of 2x i.e. (2x)2 is equal to 4x2
- When squaring a binomial, always use the correct formulae.
**E.g. Square of (2x+3) i.e. (2x+3)2 is equal to 4x2 + 9 + 12x
- When a polynomial is divided by a monomial then each term of the numerator of the polynomial is divided by the monomial present in the denominator.
\text{e.g. }\dfrac{2x+6}{3}\text{ can be written as, }\dfrac{2x}{3}+\dfrac{6}{3}\text{ and is equal to }\dfrac{2x}{3}+2.
**Conclusion
This article teaches clearly various methods to do factorization. Also, it discusses the division of algebraic expression with proper examples.
- When a monomial is divided by other monomials, the coefficient of the quotient of two monomials is equal to the quotient of their coefficients.
- When a monomial is divided by other monomials, the variable of the quotient of two monomials is equal to the quotient of the variables of monomials.
- We can check the result by **Dividend = Remainder + Divisor × Quotient
**Also, Check