Division of Algebraic Expressions

Division of algebraic expressions means simplifying one algebraic expression by another. It helps break down the complex expression to make it easier to solve. 
 

A division of algebraic expressions is used to divide one algebraic expression by another. The approach depends on the types of expressions involved, like monomial or polynomial. Understanding these types of algebraic division helps us simplify expressions correctly and solve algebraic problems.

 Types of Algebraic Division:

• Division of a Monomial by a Monomial
• Division of a Polynomial by a Monomial
• Division of a Polynomial by a Polynomial
  
   

  Types of Algebraic Division	Definition	Example
  Division of a Monomial by a Monomial	An expression has only one term.	8x3/2x = 4x2
  Division of a Polynomial by a Monomial	An expression has more than one term.	12x2 + 6x2/3x = 4x2 + 2x2
  Division of a Polynomial by a Polynomial	This method is used when dividing polynomials, mostly in long division or synthetic division	x2 + 3x + 2 / x + 1

  
   

Division of a Monomial by a Monomial is the easiest type in algebraic division, where both the dividend and divisor are monomials.

For example, divide 184 by 6x 

Solution:

Divide the coefficients first:
18 ÷ 6
= 3
Then divide the variables 
x4 ÷ x
= x4 -1
= x3
The solution is 3x³
 

Division of a Polynomial by a Monomial helps divide each term of a polynomial separately by the monomial. For example, Divide 12x2 + 6x2 by 3x
Solution:
 12x2 + 6x2 ÷  3x
Divide each of them by 3x
12x2 ÷ 3x
= 4x2.
6x2 ÷ 3x
= 2x
Combine the simplified part like terms 4x2 + 2x
 

Division of a Polynomial by a Polynomial helps to divide a polynomial by another polynomial. We cannot just divide term by term. Instead, we use a similar method to long division (like we do with numbers). This is called polynomial long division. For example, x2 + 3x + 2 / x + 1
Solution:
Determine how many times the divisor (x + 1) divides into the dividend (x² + 3x + 2).
Divide the first term x2 ÷ x = x
Multiply x with the divisor (x + 1)
x(x + 1) = x2 + x
Subtract (x2 + 3x + 2) - (x2 + x) = 2x + 2
Divide again:
2x ÷ x 
x = 2
Multiply 2(x + 1) = 2x + 2
Subtract (2x + 2) - (2x +2) = 0
The solution is x + 2
 

There are several main methods to perform the division of Algebraic expressions. They are 
 

• Long Division Method
• Synthetic Division
  
   

Long Division Method:
The long division method is a way to divide one polynomial by another, kind of like how we divide large numbers using regular long division.

For example: x2 + 3x + 2 / x + 1

Solution:

Divide the first term x2 ÷ x = x
Multiply and subtract 
Multiply x by x + 1
x(x + 1) = x2 + x
Then subtract:
(x2 + 3x + 2) - (x2 + x) = 2x + 2
Divide the new term
2x ÷ x = 2
Multiply and subtract again 
2(x + 1) = 2x + 2
(2x + 2) - (2x + 2) = 0
The solution is x + 2

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear divisor of the form x-c. For example: Divide x² + 3x + 2 by x + 1 (c = -1).
Write the coefficients: 1, 3, 2.
Synthetic division with c = -1 | 1 3 2 
                                              |    -1 -2
                                              | 1 2 0
The bottom row is 1, 2, 0. These are the coefficients of the quotient and the remainder
The answer is x + 2.
 

Apart from the division, there are other operations on algebraic expressions, including addition, subtraction, and multiplication.
Addition of algebraic expressions
Combine like terms that have the same variables and exponents.
For example:
Add (3x + 7) + (2x + 4)
Solution: combine like terms 
(3x + 2x) = 5x
(7 + 4) = 11
The answer is 5x + 11 
 

Same as addition, but distribute the minus sign, then combine like terms. For example
(5x + 3) - (2x + 6)
Solution: 
In the subtraction method, the first step is to remove the brackets.
5x + 3 -2x + 6
Combine like terms
5x -2x
= 3x
6-3
=3
The answer is 3x + 3
 

Multiplication of algebraic expressions does not require combining like terms during multiplication; like terms are combined afterward. For example, (x + 2) (x + 3).
Solution:
x  × x = x2
x  × 3 = 3x
2 × x = 2x
2  × 3 = 6 
Add all the terms:
x2 + 3x + 2x + 6
Combine like terms 
x2 + 5x + 6
The answer is x2 + 5x + 6.
 

Division of algebraic expressions isn’t only a math subject, it also helps to solve real-world problems. By dividing algebraic expressions, we can simplify complex scenarios into manageable ‘per unit’ values, such as cost per item, speed per hour, or dosage per person. The following are some real-life applications.

Computer graphics and design: The division of algebraic expressions plays a role in computer-aided design (CAD) and engineering. It helps to create and analyze the shapes of objects, like designing smooth surfaces for cars, modeling 3D objects for printing, or even recreating parts in reverse engineering.

 
Construction and Architecture: Calculating how many bricks are needed to build a wall. Knowing the total length of a wall (algebraic expression) and the length of one brick allows you to divide the two expressions to determine the number of bricks required.

Physics: Division of algebraic expressions is commonly used in physics to solve equations, which is applied in fields like electromagnetics, quantum field theory, geometric optics, and geometric mechanics.