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Explore The Basics of Synthetic Division And Its Importance In Polynomial Arithmetic

In algebra, one of the best methods to perform the Euclidean division of polynomials is through the synthetic division. It can also be done with the help of the long division but the synthetic division method requires less calculation. Meanwhile, it’s the shorter way of polynomial division. Despite sounding complex, polynomials are expressions like 2x² + 3x – 5, where ‘x’ is the variable and the numbers are coefficients. with variables and coefficients. 

Understanding how to work with polynomials is essential in various mathematical endeavours. If you don’t know how to perform the synthetic division of the polynomials, no need to worry, our synthetic division calculator can help. This tool just requires a few inputs and performs the whole calculations rapidly providing you with precise results with the steps shown. Read on! To Explore the synthetic division of polynomials with detailed steps and solved examples.

What is Synthetic Division?

This is a method that lets you perform the division operation on polynomials by a linear factor such as x + 2. In comparison to the long traditional method, the synthetic division method is much easier and shorter because it does not require writing variables while performing the polynomial division.  The division of two polynomials can be represented as follows:
p(x)/q(x) = Q(x) + R/(q(x))


  • p(x) represents the dividend
  • q(x) is the linear divisor
  • Q(x) represents the quotient
  • R is the remainder

How Do You Do Synthetic Division?

To perform the synthetic division of polynomials, follow the below-mentioned steps:

  • Write down “k” as the divisor
  • Note down the coefficients of the dividend
  • Drop the first coefficient down
  • Now, multiply the first coefficient by k. Write the result of the product in the next column
  • Add the terms of the second column
  • Multiply the result by k. Put the product in the next column
  • Continue repeating steps 5 and 6 for the remaining columns

Skip the hassle of manual calculations, go to the Online calculator, and use our division calculator to swiftly compute the quotient and remainder, streamlining your polynomial arithmetic tasks.


Divide the polynomial (4x^2 -5x -21) by the linear divisor (x – 3) using the synthetic division method.  


Coefficients of the numerator polynomial

4, -5, -21

Find the zeros of the denominator

x – 3 = 0x = 3.0

Synthetic division format

Drop down the leading coefficient to the bottom row Multiply the obtained value by k, and put the outcome into the next column

4*(3.0) = 12

Add down the column

-5 + (12) = 7

Multiply the resultant value by k, and put the outcome into the next column

7 * (3.0) = 21

Add down the column

-21 + (21) = 0

Now the quotient is 4x + 7, and the remainder is zero

The Role of Synthetic Division Calculator:

The role of this calculator is crucial in the landscape of mathematical computation, especially when dealing with polynomial division. It divides two polynomials using synthetic division to find the quotient and the remainder. This tool enhances efficiency, accuracy, and understanding of the whole calculation in different ways and serves various roles, including:

  • Simplification of Polynomial Division: The presence of an online tool makes synthetic division a breeze for those who find manual calculations challenging 
  • Time-Saving: Once you input the coefficient and divisor, the tool instantly provides you with the quotient and remainder along with the step-by-step solution. Meanwhile, it’s a handy tool for students who are studying polynomials. With its help, they can verify their results and make their assignments
  • Accuracy: The use of the online tool brings convenience in performing the division. It eliminates the chance of human error in the calculations
  • Broad Accessibility: Anyone having an internet connection can access and use the tool. This way, it provides equal learning opportunities and support to students and professionals of the whole world
  • Versatility in Applications: This tool is not just for academic purposes, it also serves in various professional fields, such as engineering, economics, and the sciences

In the world of polynomial exploration, there’s another helpful tool worth mentioning, the synthetic substitution calculator. This tool helps to substitute a specific value into a polynomial and quickly evaluate the result.

Importance of Synthetic Division in Polynomial Arithmetic:

  • It offers a streamlined method for efficient division
  • Simplify complex calculations
  • Ensures accuracy
  • Helps in tackling polynomial challenges with ease


In the wide world of mathematics, synthetic division shines as a standout method for dealing with polynomials, especially when you’re working with linear divisors. Thanks to the advancement of technology and online tools like the synthetic division calculator, which helps master these techniques easily. As you dive deeper into your math adventures, this tool turns the tricky polynomial challenges into smooth sailing.

Birm Gehlot

बिरम गहलोत एक Successful Full Time Blogger है| HelpStudentPoint.com के Founder और Senior Editor In Chief है| उन्होंने MCA की पढ़ाई पूरी की है| इन्होने Blogging Career की शुरुआत 2016 में किया था और अभी तक कई सारे बिज़नेस और सक्सेसफुल ब्लॉग बना चुके है| बिरम गहलोत से संबंधित अधिक जानकारी के लिए यहां क्लिक करें


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