# Division of Algebraic Expressions

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 mini-lesson, 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

 1 How Do You Divide Algebraic Expressions? 2 Important Notes 3 Solved Examples 4 Challenging Questions 5 Interactive Questions

## 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 n-terms.

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.

### 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?

In the division of two algebraic expressions, we cancel the common terms from the expressions, which is similar to the division of the numbers.

### 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.

More Important Topics
Numbers
Algebra
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Money
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Trigonometry
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