Division of Algebraic Expressions
Emily recently learned about algebraic expressions and now she is wondering if she can divide those expressions in the same way as she divides the two whole numbers or fractions together.
Do you want to help Emily with the division of algebraic expressions?
Let's find out dividing polynomials or algebraic expressions.
In this minilesson, we will explore the division of algebraic expressions by learning about like and unlike terms, methods to dividing polynomials with the help of interesting simulation, some division of algebraic expressions worksheets, some solved examples, and a few interactive questions for you to test your understanding
Lesson Plan
What Do You Mean by Like and Unlike Terms?
Like Terms
In algebra, the terms that contain the same variable which is raised to the same power are called like terms. In the like terms, the numerical coefficients can vary.
Like terms are combined to simplify the algebraic expressions due to which the result can be obtained very easily for the expression.
For example,
2y + 12y contains like terms.
For further simplification, this algebraic expression can be added directly because these are the like terms. Thus, the simplification of the given expression is 14y. Similarly, we can perform all the arithmetic operations on the like terms.
Unlike Terms
In algebra, the terms that do not have the same literal coefficients, and their power cannot be raised as same are called, unlike terms.
For example,
3y + 5x contains unlike terms.
For further simplification, this algebraic expression can not be added directly because these are the unlike terms, it has two different variables x and y, and not raised to the same power.
How Do You Divide Algebraic Expressions?
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.
In the division of an algebraic expression, we cancel the common terms, which is similar to the division of the numbers.
\[27x^3 ÷ 3x = \dfrac{27 \times x \times x \times x}{ 3 \times x }\]
Now, cancel out the common terms, we get:
\[27x^3 ÷ 3x = 9x^2\]
Division of Polynomial by a Monomial
A polynomial contains a few types of expressions, some of which are a 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 monomial is 2y.
In trinomial, on taking the common factor 2y, it becomes:
\[4y^3 + 5y^2 + 6y = 2y (2y^2 + \left(\dfrac52\right)y + 3)\]
Now, we do the division operation:
\[\{2y (2y^2 +\left (\dfrac52\right)y + 3)\}\div 2y\]
On canceling the 2y from the numerator and the denominator, it becomes:
\[(4y^3 + 5y^2 + 6y) ÷ 2y = 2y^2 + \left(\dfrac52\right)y + 3\]
Division of Polynomial by a Polynomial
Let us consider polynomials divides polynomial for performing the division operation.
\[(7x^2 + 14x) ÷ (x + 2)\]
Here, both polynomials exist in the binomial form. Similar to the above process, we will first 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,
\[7x^2 + 14x = 7x(x+2)\]
Let's now do the division of algebraic operation,
\[(7x^2 + 14x) ÷ (x + 2) = \dfrac{7x(x+2)}{ (x+2)}\]
On eliminating (x+2) from the numerator and denominator, we get the solution for the long dividing polynomials as:
\[(7x^2 + 14x) ÷ (x + 2) = 7x\]
\(\therefore \) The solution for the division is 7x. 
 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.
Solved Examples
Example 1 
Sam is confused about the like and unlike terms in the algebraic expression \(8x^2y + 3xy^2 – xy – 7yx^2\). Can you help him find out?
Solution
Here, the like terms are \(8x^2y, – 7yx^2\), each term is having the same literal coefficients \(x^2y\).
And the unlike terms are \(3xy^2, – xy\), each term is having different literal coefficients.
Like terms: \(8x^2y, – 7yx^2\) Unlike terms: \(3xy^2, – xy\) 
Example 2 
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:
\[4x^3  3x^2 + 4x = 2x (2x^2  \left(\dfrac32\right)x + 2)\]
Now, we divide the expression by 2x:
\[(4x^3  3x^2 + 4x) ÷ 2x = \{2x (2x^2  \left(\dfrac32\right)x + 2)\} \div 2x\]
On cancelling the 2x from numerator and the denominator, we get:
\[2x^2  \left(\dfrac32\right)x + 2\]
\( \therefore (4x^3  3x^2 + 4x) \div 2x = 2x^2  \left(\dfrac32\right)x + 2\) 
Example 3 
Can we help Abigail 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,
\[(24a^2 + 48a) ÷ (6a + 12) = [4a(6a+12)] \div (6a+12)\]
On eliminating (6a+12) from the numerator and denominator, we get the solution for the division.
Thus, \( [4a(6a+12)] \div (6a+12) = 4a\)
\(\therefore (24a^2 + 48a) ÷ (6a + 12) = 4a\) 
 Divide \((2x³ – 5x² – x + 7)\) by \((4x² 2x +3)\).
 Divide \(60y^4 + 22y^3 − 164y^2 − 24y + 84\) by \(6y^2 + 4y− 8\) and find the quotient and remainder of the expression.
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The minilesson targeted the fascinating concept of division of algebraic expression and dividing polynomials examples. The math journey around the division of algebraic expression started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.
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FAQs on Multiplication of Algebraic Expressions
1. How do you classify expressions?
We can classify the algebraic expressions into the following five categories.
They are monomial, polynomial, binomial, trinomial, and multinomial.
2. How do you write an algebraic expression?
We can write algebraic expressions using variables, constants, and, coefficients. For example 6x+2=14. Here "x" is a variable and 6 is its coefficient and 2 and 14 are constants.
3.What are the parts of an algebraic expression?
An algebraic expression has different parts like constants, terms, like terms, coefficients, etc.
4. How can you simplify algebraic expressions?
The basic steps to simplify an algebraic expression:
1. Remove parentheses by multiplying the factors.
2. Use the exponent rule to remove parentheses in terms of exponents.
3. Combine like terms by adding their coefficients.
4. Combine all the constants.
5. How do you divide algebraic expressions?
In the division of an algebraic expression, we cancel the common terms, which is similar to the division of the numbers.
6. Can you divide like terms?
Yes, we can divide like terms.
7. How do you divide, unlike terms?
We can divide unlike terms by taking common terms from the expressions.
8. What are the steps in multiplying algebraic expressions?
The steps in multiplying algebraic expressions are:
1. Take out the factor from both the denominator and numerator of each expression.
2. Reduce the expressions to the lowest terms possible only if taking out common terms.
3. Multiply together all the remaining expressions.
9. How do you divide two algebraic expressions?
10. How do you divide rational algebraic expressions?
Few steps to divide rational algebraic expressions are:
1: Make factors of both the numerators and denominators of all fractions.
2: Change the division sign into a multiplication sign and reciprocate the fraction and further multiply the terms.
3: Cancel or reduce the fractions.