Division of Algebraic Expressions(With Methods & Examples)

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In mathematics, addition, subtraction, multiplication, and division are four basic operations. Just like we divide numbers we can perform the division of algebraic expressions.  In the case of addition and subtraction, we can add or subtract only the like terms. But in the case of the division of algebraic expressions, participating terms don’t need to be the like terms. We can divide two or more like terms as well as unlike terms.

Let’s understand the methods of dividing algebraic expressions with steps and examples.

What is the Division of Algebraic Expressions?

The division of algebraic expressions is a method of dividing two given expressions consisting of variables and constants. The general procedure involved in the division of algebraic expressions is to 

• divide the coefficients of the terms
• subtract the powers of the variables with the same base
• obtain the algebraic sum of the like and unlike terms

For example, $32x^{6} \div 4x^{2} = \left(32 \div 4 \right) \left(x^{6} \div x^{2} \right) = 8x^{6 – 2} = 8x^4$

While dividing the coefficients, the general rules of division of integers are followed.

• $\text{Positive} \div \text{Positive} = \text{Positive}$, i.e., $+ \div + = +$
• $\text{Positive} \div \text{Negative} = \text{Negative}$, i.e., $+ \div – = -$
• $\text{Negative} \div \text{Positive} = \text{Negative}$, i.e., $- \div + = -$
• $\text{Negative} \div \text{Negative} = \text{Positive}$, i.e., $- \div – = +$

And, while dividing variables, the division rule of exponents is followed

• $a^{m} \div a^{n} = a^{m – n}$

How to Divide Algebraic Expressions?

Algebraic expressions are broadly classified into two types – monomials and polynomials. Depending on the type of algebraic expressions these are the three different types of division of algebraic expressions.

• Dividing a monomial by a monomial
• Dividing a polynomial by a monomial
• Dividing a polynomial by a polynomial

Dividing a Monomial by a Monomial

An algebraic expression is considered a monomial when it contains only one term, such as $2x^{3}$, $b^{4}$, etc. While dividing a monomial by another monomial, the quotient of the coefficient of the two monomials is calculated and the variables are divided separately.

Let’s consider a few examples to understand the procedure of dividing a monomial by a monomial.

Examples

Ex 1: $15m^{5} \div 3m^{2}$

$15m^{5} \div 3m^{2} = \frac{15m^{5}}{3m^{2}} = \frac{15}{3} \times \frac{m^{5}}{m^{2}} = 5m^{5 – 2} = 5m^{3}$

Ex 2: $12x^{3}y^{5} \div 5xy^{2}$

$12x^{3}y^{5} \div 5xy^{2} = \frac{12}{5} \times \frac{x^{3}y^{5}}{xy^{2}} = \frac{12}{5} \times \frac{x^{3}y^{5}}{xy^{2}} = \frac{12}{5}x^{3 – 1}y^{5 – 2} = \frac{12}{5}x^{2}y^{3}$

Ex 3: $8bx^{2}y \div 2axy^{2}$

$8bx^{2}y \div 2axy^{2} = \frac {8}{2} \times \frac{bx^{2}y}{axy^{2}} = 4 \frac{bx}{ay}$

Dividing a Polynomial by a Monomial

There are many types of polynomials depending on the number of terms they contain such as 

• binomial having two terms
• trinomial having three terms
• polynomial having more than three terms

To simplify these types of expressions, we look for common factors. A common factor is found when we have the same number or variable or a combination of number and variable in the numerator and denominator.

Examples

Now, let’s perform the division of polynomials by monomials.

Ex 1: $\left(4y^{3} – 6y^{2} + 7y \right) \div 2y$

Here, the trinomial is $4y^{3} – 6y^{2} + 7y$, and the monomial is $2y$.

In trinomial, on taking the common factor $2y$, it becomes: 

$4y^{3} – 6y^{2} + 7y = 2y \left(2y^{2} – 3y + \frac{7}{2} \right)$

Now, we do the division operation: $2y \left(2y^{2} – 3y + \frac{7}{2} \right) \div 2y = \frac{2y \left(2y^{2} – 3y + \frac{7}{2} \right)}{2y}$.

Cancel $2y$ from the numerator and the denominator, so we get $2y^{2} – 3y + \frac{7}{2}$

Thus, $\left(4y^{3} – 6y^{2} + 7y \right) \div 2y = 2y^{2} – 3y + \frac{7}{2}$.

Ex 2: $\left(2ay^{4} + 6by^{2} + 4aby \right) \div 2ab$

Here, the trinomial is $2ay^{4} + 6by^{2} + 4aby$, and the monomial is $2ab$.

In trinomial, on taking the common factor $2y$, it becomes: 

$2ay^{4} + 6by^{2} + 4aby = 2y\left(ay^{3} + 3by + 2ab \right)$

Now, we do the division operation: $\left(2ay^{4} + 6by^{2} + 4aby \right) \div 2ab = \frac {2y\left(ay^{3} + 3by + 2ab \right)}{2y}$.

Cancel $2y$ from the numerator and the denominator, so we get $ay^{3} + 3by + 2ab$

Thus, $\left(2ay^{4} + 6by^{2} + 4aby \right) \div 2ab = ay^{3} + 3by + 2ab$.

Dividing a Polynomial by a Polynomial

While dividing a polynomial by a polynomial, common factors from both polynomials are taken and then the common factor is canceled out to get the quotient.

The steps to find the quotient when a polynomial is divided by another polynomial are

Step 1: Take the common factor from both the polynomials.

Step 2: Cancel out the common factor.

Step 3: The quotient is the remaining polynomial.

Examples

Let us consider polynomials that divide polynomial for performing the division operation.

Ex 1: $\left(3x^{2} + 6x \right) \div (x + 2)$

Here, both the polynomials are binomials.

Taking out the common factor from $3x^{2} + 6x$.

$3x^{2} + 6x = 3x \left(x + 2 \right)$.

Therefore, $\left(3x^{2} + 6x \right) \div \left(x + 2 \right) = \frac{3x^{2} + 6x}{x + 2} = \frac{3x \left(x + 2 \right)}{x + 2} = 3x$.

Ex 2: $\left(6x^{2} + 8x + 2 \right) \div \left(2x + 2 \right)$

Here, one polynomial is a trinomial and the other is a binomial.

Take out the common factors. 

For the polynomial $6x^{2} + 8x + 2$, $2x + 2$ is the common factor.

So, we get $6x^{2} + 8x + 2 = \left(2x +2 \right) \left(3x + 1 \right)$

Now, consider $2x + 2$ as a common factor among them. 

Therefore, $\left(6x^{2} + 8x + 2 \right) \div \left(2x + 2 \right) = \frac{\left(2x +2 \right) \left(3x + 1 \right)}{2x + 2}$.

Eliminate $2x + 2$ from the numerator and denominator, we get the solution for the long dividing polynomials as: 

$\left(6x^{2} + 8x + 2 \right) \div \left(2x + 2 \right) = 3x + 1$.

Tips for Division of Algebraic Expressions

• We can divide any algebraic term with any other algebraic term. It can be the division of two like terms or a division of like and unlike terms.
• We can ignore the order of variables in like terms in an algebraic expression. For example,  $3a + 2b$, and, $9b + a$ both are like terms.
• We can ignore writing $1$ as the numerical coefficient of any term. For example, $xy$ is the same as $1xy$.
• We can replace a missing term with $0$ with the same variables. For example, a missing term can be written as $0x$, $0y$, or $0xy$ depending on the variables of the missing term.

Practice Problems

Divide the following

• $9x^{2}$ by $3x$
• $7x^{5}$ by $2x^{2}$
• $12a^{2}b^{3}$ by $2ab$
• $15m^{2} + 5m$ by $5m + 1$
• $26a^{2} + 12a$ by $13a + 6$
• $15x^{2} + 11x + 2$ by $6x^{2} + 11x + 3$

FAQs

Conclusion

The division of algebraic expressions is a method of dividing two given expressions consisting of variables and constants. The general procedure involved in the division of algebraic expressions is to look for a common factor, which is then canceled out to get the quotient.

Recommended Reading

• Multiplication of Algebraic Expressions(With Methods & Examples)
• Subtraction of Algebraic Expressions(With Methods & Examples)
• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebraic Expression(Definition, Formulas & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)
• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)