Dividing Polynomials

Polynomials are algebraic expressions that consist of variables and constants. Polynomials can be written in the form: \(ax^2 + bx + c\), arranged in descending order of their degree.


Division is one of the basic arithmetic operations, and in algebra, it involves breaking down a polynomial into equal or simpler parts. Dividing polynomials includes dividing a polynomial by a monomial or a binomial. For example, when dividing \(\frac{2x^2 + 4x + 24}{2x + 12}\), it can be written as: 

 

\(\frac{2x^2 + 4x + 24}{2x + 12}\)

Here, the numerator is \(2x^2 + 4x + 12\) and the denominator is \(2x + 12\). That means the numerator becomes the dividend and the denominator becomes the divisor

Polynomials have multiple terms, while monomials consist of only one term. To divide a polynomial by a monomial, there are two ways: 

 

• Splitting the terms method 
  
   
• Factorization method. 

In the splitting the term method, the terms of the polynomials are split by the operations between them, and then each term is separately divided by the divisor.


For example, 22x² + 12 by 2x

 

• Splitting the term: 22x² + 12 
  
  
    The terms are 22x² and 12
   

• Dividing each term by the divisor:
  
  
   22x² / 2x = 11x

• Simplifying:
  
  
  12/2x = 6/x
  
  
  \(\frac{22x^2 + 12}{2x} = 11x + \frac{6}{x} \)

In the factorization method, we find the common factor between the numerator and denominator of the polynomial by factoring the polynomial.


For example, when dividing \(\frac{22x^2 + 12x}{2x} = 11x + 6\)

 

• The common factor from 
  
  
  \(22x^2 + 12x = 2x(11x + 6)\)

• So, it can be expressed as:
  
  
  \(\frac{2x(11x + 6)}{2x} = 11x + 6\)
  
   
• Cancelling out the common factors, here the common factor is 2x

So, \(\frac{2x(11x + 6)}{2x} = 11x + 6\)

To divide polynomials by binomials, we use the long division and synthetic division methods. We use these methods when the polynomials won’t share a common factor. 

The long division method is used to divide a polynomial by another polynomial. So, both the dividend and divisor have two or more terms. Follow these steps to divide polynomials using long division, using an example:


Step 1: Dividing the first term of the dividend by the first term of the divisor.


The result is the first term of the quotient. 


Step 2: Multiply the divisor by the answer in step 1, and write below the dividend.




Step 3: Subtract the new polynomial from the dividend.



 

Step 4: The process is repeated with the same polynomial

 
For example, when dividing \({3x^2 + 8x + 4}\) by x + 2


Dividing the first terms: \(\frac{3x^2}{x} = 3x\)


Here, 3x is the first term in the quotient. 

 
Multiply the divider (x + 2) by 3x: :\(3x(x + 2) = 3x^2 + 6x\)

Subtracting: \((3x^2 + 8x + 4) - (3x^2 + 6x) = 2x + 4\)

Divide: 2x / x = 2

Add +2 as the quotient

Multiplying (x + 2) by 2, 2x + 4

Subtracting (2x + 4) - (2x + 4) = 0

So, the quotient is 3x + 2.

The synthetic division is the method used to divide polynomials by a binomial of the form x - k. Here, the focus is on the coefficient, which makes this process quicker and easier. Follow these steps for dividing polynomials using the synthetic division:



Step 1: Find the value of k and write it on the left side


To find the value of k, we first write the divisor in the form x - k. 

 

Step 2: Writing the coefficients of the dividend on the right of K

The coefficients are written on the right and k on the left. 
 

Step 3: Bring down the coefficient


Bringing down the coefficient of the highest degree term of the dividend.

Step 4: Multiply and add


Now we multiply the k by the first coefficient and write the product below the second coefficient, and add them.


Step 5: The process is repeated


Now we multiply k by the second coefficient obtained in step 4.

Step 6: The final answer will be one degree less than the dividend. For example, if the dividend has x² then the quotient will be x. 

For example, dividing \({x^2 + 5x + 6}{\text { by }}{x - 2} \)

Here, the divisor is x - 2, so k = 2.

The dividend is: \(x^2 + 5x + 6\).

So, the coefficients are: 1, 5, 6

The coefficients are written on the right and k on the left. 

Bringing down the coefficient of the highest degree term of the dividend, here it is 1. 

Here, the value of k is 2 and the first coefficient is 1, so 2 × 1 = 2

Adding 5 + 2 = 7

Here, multiply 2 and 7, \(2 × 7 = 14\).

Write 14 below 6 and add them; \(6 + 14 = 20\)
 

Here, the highest degree of dividend is x², so the quotient's higher degree would be x. The quotient is x + 7, and the remainder is 20. 

So, \(x^2 + 5x + \frac{6}{x} - 2 = x + 7 + \frac{20}{x -2} \)

Dividing polynomials may seem tricky at first, but with the right strategies, it becomes much easier. These tips and tricks will help you solve problems step by step, avoid common mistakes, and build confidence in algebra.
 

• Dividing polynomials may seem challenging at first, but it becomes easier if you are confident with long division and comfortable working with variables. Make sure you can combine like terms and understand powers of x before starting polynomial division.
   
• To divide polynomials, you take the first term of the dividend and divide it by the first term of the divisor, then multiply the divisor by this result and subtract it from the dividend. After that, bring down the next term and keep repeating until finished. Writing everything clearly and keeping the terms lined up helps avoid mistakes.
   
• Remainders are normal when the divisor doesn’t fit perfectly. Learn to write them as fractions over the divisor, and check your work by multiplying the quotient by the divisor and adding the remainder to see if it equals the original polynomial.
   
• Synthetic division is a useful shortcut, but only after mastering long division. Focus on understanding the logic first, then move to faster methods.
   
• Parents can help students understand how the steps work rather than memorizing them. Ask them to explain how to divide polynomials out loud as they work through a problem.
   
• Teachers can encourage neat organization when teaching long division. Lining up like terms is essential for success in the long division of polynomials.
   
• Teachers can provide dividing polynomials worksheets to identify patterns in mistakes, such as incorrect subtraction or skipped steps.

The division of polynomials is used in different fields such as engineering, computer graphics, economics, civil engineering, and so on. Here are some applications of dividing polynomials. 
 

• Engineering and Construction: Dividing polynomials is helpful in engineering and construction. It can show how forces, sizes, or materials behave. For example, to find how a beam carries weight, engineers can divide the total load (a polynomial) by the beam’s length. This helps them understand how the weight spreads along the beam.
   
• Physics: In physics, motion, energy, and paths are often shown using polynomials. Dividing polynomials makes it easier to find things like speed, acceleration, or distance per time. For example, dividing a position polynomial by time gives the average speed.
   
• Computer Graphics and Animation: In computer graphics and animation, curves and surfaces are often represented by polynomials. Dividing polynomials aids in scaling, interpolating, or simplifying these shapes. For example, adjusting a Bézier curve in animation requires dividing its polynomial representation to achieve smooth motion paths.
   
• Astronomy and Space Science: Astronomers use polynomial equations to model planetary motion, satellite orbits, or light intensity variations. Dividing polynomials helps determine average speeds, orbital distances, or time intervals between events. For instance, dividing a polynomial representing the position of a planet by time gives its average velocity in orbit.
   
• Robotics: In robotics, polynomials are used to show how robot parts move. Dividing these polynomials helps find average speed or time for a movement. For example, if a robot arm’s position is given by a polynomial, and it must finish a task in a certain time, dividing by the time shows the average speed needed. This helps robots move smoothly and do tasks like picking, placing, or assembling objects.