Division of Algebraic Expressions

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 24y³ 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(m²np+ mn²p+mnp²) 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: (12m² + 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)= 12m²+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 m⁴-m³+m² +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 m⁴ - n⁴ by m² - n²

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: (18y³) / (2y²) when y = 2

Q2. Evaluate: (40p⁴q²) / (5p^2q) when p = 2 and q = 3

Q3. Divide the following polynomials and find the quotient and remainder: (10x² - 7x + 8) by (5x - 35x - 3)

Q4. Determine whether the first polynomial is a factor of the second: (x + 1) in (2x² + 5x + 4)

Q5. Divide the following polynomials and find the quotient and remainder: (3x² + 4x + 5) by (x - 2)

Q6. Divide the following polynomials and find the quotient and remainder: (x⁴ - x³ + 5x) by (x - 1)

Q7. Is (x + 1) a factor of (2x² + 5x + 4)?

Q8. Is (2a - 3) a factor of (10a² - 9a - 5)?

Q9. Write the degree of 2x + x² – 8

Q10. Evaluate: (72a⁵b³c) / (12a³b²) 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.

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

• Dividing Polynomials
• Long Division Method
• Factorization of Algebraic Expression