103 LearnersLast updated on June 24th, 2025
Synthetic Division Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like polynomial division. Whether you’re working on algebra assignments, engineering calculations, or data analysis, calculators will make your life easier. In this topic, we are going to talk about synthetic division calculators.
What is Synthetic Division Calculator?
A synthetic division calculator is a tool to perform division of polynomials quickly and efficiently. It simplifies the traditional long division method by only focusing on the coefficients, making the process faster and more straightforward.
This calculator helps you divide polynomials, especially when dividing by a linear factor, saving time and effort.
How to Use the Synthetic Division Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the coefficients: Input the coefficients of the polynomial you want to divide into the given fields.
Step 2: Enter the divisor: Input the linear factor's root (the divisor) into the appropriate field.
Step 3: Click on calculate: Click on the calculate button to perform the division and get the result.
Step 4: View the result: The calculator will display the quotient and remainder instantly.
How to Perform Synthetic Division?
To perform synthetic division, you follow a specific process using the coefficients of the polynomial. Synthetic division is typically used for dividing a polynomial by a linear factor of the form (x - c). Here's the basic idea:
1. Write down the coefficients of the polynomial.
2. Use the root (c) of the divisor (x - c) for the synthetic division process.
3. Bring down the leading coefficient.
4. Multiply and add repeatedly to find the new coefficients of the quotient and the remainder.
The result will show you the quotient and the remainder of the division.
Tips and Tricks for Using the Synthetic Division Calculator
When we use a synthetic division calculator, there are a few tips and tricks that we can use to make it a bit easier and avoid mistakes:
- Understand the polynomial's degree and the divisor's degree to know the expected degree of the quotient.
- Ensure that all coefficients, including zero coefficients, are included for missing terms.
- Verify if the divisor is in the form (x - c) to use synthetic division effectively.
- Use the calculator to check manual calculations for accuracy.
Common Mistakes and How to Avoid Them When Using the Synthetic Division Calculator
We may think that when using a calculator, mistakes will not happen. But it is possible for users to make mistakes when using a calculator.
Mistake 1
Forgetting to include zero coefficients for missing terms.
Make sure to include all coefficients, even if a term is missing in the polynomial, to avoid incorrect results.
Mistake 2
Using the wrong root for the divisor.
Ensure that the divisor is in the form (x - c), and use c as the number for synthetic division.
Mistake 3
Confusing the remainder with the quotient.
Remember that the remainder is the last value after the synthetic division process and is separate from the quotient.
Mistake 4
Relying on the calculator without understanding the process.
Understanding synthetic division will help you interpret the results and verify the accuracy of the calculator's output.
Mistake 5
Assuming the calculator will handle all polynomial divisions.
Synthetic division is only applicable for division by linear factors. For more complex divisors, use traditional polynomial division.
Synthetic Division Calculator Examples
Problem 1
Divide the polynomial 2x³ + 3x² - 5x + 6 by x - 2.
Perform synthetic division using the coefficients [2, 3, -5, 6] and the root c = 2:
1. Bring down the 2.
2. Multiply 2 by 2 and add to the next coefficient: 3 + 4 = 7.
3. Multiply 7 by 2 and add to the next coefficient: -5 + 14 = 9.
4. Multiply 9 by 2 and add to the last coefficient: 6 + 18 = 24.
Quotient: 2x2 + 7x + 9
Remainder: 24
Explanation
By using synthetic division, you transform the polynomial division into simpler arithmetic steps, resulting in a quotient of 2x2 + 7x + 9 and a remainder of 24.
Problem 2
Divide the polynomial x^3 - 6x^2 + 11x - 6 by x - 1.
Perform synthetic division using the coefficients [1, -6, 11, -6] and the root c = 1:
1. Bring down the 1.
2. Multiply 1 by 1 and add to the next coefficient: -6 + 1 = -5.
3. Multiply -5 by 1 and add to the next coefficient: 11 - 5 = 6.
4. Multiply 6 by 1 and add to the last coefficient: -6 + 6 = 0.
Quotient: x2 - 5x + 6
Remainder: 0
Explanation
Synthetic division yields a quotient of x2 - 5x + 6 with a remainder of 0, indicating the divisor is a factor of the polynomial.
Problem 3
Divide the polynomial 3x² - 12x² + 9x² + x - 4 by x + 3.
Perform synthetic division using the coefficients [3, -12, 9, 1, -4] and the root c = -3:
1. Bring down the 3.
2. Multiply 3 by -3 and add to the next coefficient: -12 - 9 = -21.
3. Multiply -21 by -3 and add to the next coefficient: 9 + 63 = 72.
4. Multiply 72 by -3 and add to the next coefficient: 1 - 216 = -215.
5. Multiply -215 by -3 and add to the last coefficient: -4 + 645 = 641.
Quotient: 3x3 - 21x2 + 72x - 215
Remainder: 641
Explanation
By following the synthetic division steps, you obtain a quotient of 3x3 - 21x2 + 72x - 215 with a remainder of 641.
Problem 4
Divide the polynomial 4x² - 2x + 1 by x - 1.
Perform synthetic division using the coefficients [4, -2, 1] and the root c = 1:
1. Bring down the 4.
2. Multiply 4 by 1 and add to the next coefficient: -2 + 4 = 2.
3. Multiply 2 by 1 and add to the last coefficient: 1 + 2 = 3.
Quotient: 4x + 2
Remainder: 3
Explanation
The synthetic division provides a quotient of 4x + 2 with a remainder of 3.
Problem 5
Divide the polynomial 5x³ + 0x² - 4x + 7 by x - 2.
Perform synthetic division using the coefficients [5, 0, -4, 7] and the root c = 2:
1. Bring down the 5.
2. Multiply 5 by 2 and add to the next coefficient: 0 + 10 = 10.
3. Multiply 10 by 2 and add to the next coefficient: -4 + 20 = 16.
4. Multiply 16 by 2 and add to the last coefficient: 7 + 32 = 39.
Quotient: 5x2 + 10x + 16
Remainder: 39
Explanation
By using synthetic division with the root 2, the quotient is 5x2 + 10x + 16, and the remainder is 39.
FAQs on Using the Synthetic Division Calculator
1.How do you calculate synthetic division?
2.Can synthetic division be used for all polynomial divisions?
3.What is the main advantage of synthetic division?
4.Can I use synthetic division for complex numbers?
5.Is the synthetic division calculator accurate?
Glossary of Terms for the Synthetic Division Calculator
- Synthetic Division: A simplified method of dividing polynomials using coefficients.
- Coefficients: The numerical factors in a polynomial's terms.
- Quotient: The result obtained from the division process.
- Remainder: The value left over after division.
- Linear Factor: A polynomial of the form (x - c), where c is a constant.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
