Synthetic Division Calculator – Quick Polynomial Division Tool
Polynomial division can be tricky, especially when dealing with long coefficients and signs. That’s where the Synthetic Division Calculator comes in — a simple yet powerful tool that helps you divide a polynomial by a binomial of the form (x – r) quickly and accurately. Whether you’re a student learning algebra or someone working through higher math problems, this calculator saves time and reduces calculation errors.
🟣 Synthetic Division Calculator
Divide a polynomial by a binomial using synthetic division.
How to Use the Synthetic Division Calculator
Here’s how to perform a division using the calculator:
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Enter the Polynomial Coefficients
In the Dividend box, type all coefficients of your polynomial, separated by commas.
Example:represents the polynomial:
2x³ – 6x² + 2x – 1 -
Enter the Divisor Value (r)
The divisor must be in the form x – r.
For example, if dividing by x – 3, enter 3 as your divisor value. -
Choose Output Format
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Decimal: Displays results in decimal form (useful for non-integer results).
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Fraction: Shows precise fractional coefficients (best for exact math work).
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Click “Calculate”
The calculator will instantly perform synthetic division and display the quotient and remainder, step by step.
What Is Synthetic Division?
Synthetic division is a simplified way to divide a polynomial by a binomial of the form (x – r). It’s faster and easier than long division because it uses only coefficients — no variable manipulation needed!
Instead of writing out each term, you perform arithmetic with the coefficients directly.
For example, dividing:
2×3−6×2+2x−1 by (x−3)2x^3 – 6x^2 + 2x – 1 \text{ by } (x – 3)
results in a quotient of
2×2+0x+2 and a remainder of 52x^2 + 0x + 2 \text{ and a remainder of } 5
Formula Used in Synthetic Division
The process follows this pattern:
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Write the coefficients of the polynomial in order.
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Bring down the first coefficient as is.
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Multiply it by the divisor root (r) and add to the next coefficient.
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Repeat until you reach the end.
This gives you new coefficients for the quotient, with the final number being the remainder.
For example:
| Step | Coefficient | Multiply by r | Add | New Value |
|---|---|---|---|---|
| 1 | 2 | – | – | 2 |
| 2 | -6 | (2×3)=6 | -6+6=0 | 0 |
| 3 | 2 | (0×3)=0 | 2+0=2 | 2 |
| 4 | -1 | (2×3)=6 | -1+6=5 | 5 (Remainder) |
Result:
Quotient: 2×2+0x+2,Remainder: 5\text{Quotient: } 2x^2 + 0x + 2,\quad \text{Remainder: } 5
Example Calculations
Example 1:
Divide x³ – 5x² + 2x + 8 by x – 2
Input:
Dividend → 1, -5, 2, 8
Divisor (r) → 2
Result:
Quotient = x² – 3x – 4
Remainder = 0
✅ This means (x – 2) is a factor of the polynomial!
Example 2:
Divide 2x³ + 3x² – 11x + 6 by x – 2
Input:
Dividend → 2, 3, -11, 6
Divisor (r) → 2
Output:
Quotient = 2x² + 7x + 3
Remainder = 12
Why Use Synthetic Division?
Synthetic division is faster than long division and helps you:
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✅ Find factors of polynomials.
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✅ Check roots or zeros (if remainder = 0).
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✅ Simplify high-degree equations.
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✅ Prepare for graphing or factorization.
It’s commonly used in algebra, pre-calculus, and calculus when solving polynomial functions or simplifying equations.
Long Division vs. Synthetic Division
| Feature | Long Division | Synthetic Division |
|---|---|---|
| Works with | Any polynomial divisor | Only divisors of form (x – r) |
| Uses variables | Yes | No |
| Time to solve | Longer | Shorter |
| Best for | General equations | Simple divisors |
| Complexity | High | Low |
So if your divisor is (x – r) — synthetic division is the fastest, most efficient choice!
Fraction vs. Decimal Output
The calculator lets you choose between Fraction and Decimal formats:
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Fraction: Ideal for exact results, such as
3/2instead of1.5. -
Decimal: Better when approximations are acceptable or needed for quick checks.
Behind the Math
Let’s say your polynomial is:
anxn+an−1xn−1+…+a1x+a0a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0
and you divide by (x−r)(x – r).
Each step replaces the coefficient as:
bi=ai+(bi−1×r)b_i = a_i + (b_{i-1} \times r)
The final term bnb_n gives the remainder, and the rest form the quotient coefficients.
Educational Applications
Synthetic division is taught in high school and early college algebra. It’s used in:
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Factoring polynomials to find roots.
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Checking for zeroes of polynomial functions.
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Building graphs for cubic and quartic functions.
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Finding asymptotes and intercepts in calculus.
By using this calculator, students can check their manual work instantly.
Example for Fraction Output
Divide x² – 3x + 2 by x – 4
Input:
Dividend → 1, -3, 2
Divisor → 4
Result:
Quotient = x + 1
Remainder = 6
In fraction format, the remainder can be shown as:
6x−4\frac{6}{x – 4}
So,
x2−3x+2x−4=x+1+6x−4\frac{x^2 – 3x + 2}{x – 4} = x + 1 + \frac{6}{x – 4}
Conclusion
The Synthetic Division Calculator is your perfect companion for fast, accurate, and step-by-step polynomial division. It’s built for students, teachers, and professionals who want to simplify complex algebraic expressions in seconds.
✅ No more errors.
✅ No more tedious work.
✅ Just enter coefficients and get instant results — in fraction or decimal form.
Master algebra the smart way with the Synthetic Division Calculator — where math meets simplicity!
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