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Synthetic Division Calculator
Divide a polynomial by x − c, by a linear factor like 2x + 1, or by any polynomial. Linear divisors get the classic synthetic division table; higher-degree divisors are solved with polynomial long division. Every step uses exact fractions.
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Expression (x^3 - 2x + 1) or coefficients from the highest power (1, 0, -2, 1).
x − c, ax + b, or a higher-degree polynomial like x^2 + 1
Enter your values to see the result.
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How to use the Synthetic Division Calculator
- Enter the dividend, for example 2x^4 − 3x^3 + 5x − 7, or its coefficients.
- Enter the divisor: x − 2, 2x + 1, x^2 + 1, or just the number c for x − c.
- Press Divide.
- Read the quotient and remainder and the synthetic table or long division steps.
- Copy the results for your homework or notes.
Synthetic division example
Divide 2x⁴ − 3x³ + 0x² + 5x − 7 by x − 2 (so c = 2):
| Step | 2 | −3 | 0 | 5 | −7 |
|---|---|---|---|---|---|
| Multiply by 2 | 4 | 2 | 4 | 18 | |
| Add | 2 | 1 | 2 | 9 | 11 |
Quotient: 2x³ + x² + 2x + 9, remainder 11. So 2x⁴ − 3x³ + 5x − 7 = (x − 2)(2x³ + x² + 2x + 9) + 11, and by the remainder theorem p(2) = 11.
Division rules at a glance
| Divisor | Method | Note |
|---|---|---|
| x − c | Synthetic division with c | Remainder = p(c) |
| x + c | Synthetic division with −c | x + 3 means c = −3 |
| ax + b | Synthetic division with −b/a, then divide the quotient by a | Remainder unchanged |
| Degree 2 or more | Polynomial long division | Remainder degree < divisor degree |
Always: dividend = divisor × quotient + remainder, and the quotient's degree is the dividend's degree minus the divisor's degree.
Frequently asked questions
How does synthetic division work?
Write c from the divisor x − c and the dividend's coefficients. Bring down the first coefficient, multiply it by c, add it to the next coefficient, and repeat. The last number is the remainder and the others are the coefficients of the quotient, one degree lower than the dividend.
What if a power of x is missing?
Use a 0 for every missing power. For x³ − 2x + 1 the coefficients are 1, 0, −2, 1. This calculator fills in the zeros automatically when you type an expression.
Can I use synthetic division with 2x + 1?
Yes. Use c = −1/2 (the root of 2x + 1), do the synthetic division, then divide the resulting quotient coefficients by 2. The remainder stays the same. The calculator does this scaling for you.
What is the remainder theorem?
When a polynomial p(x) is divided by x − c, the remainder equals p(c). So synthetic division is also a fast way to evaluate a polynomial at a point.
How do I know if x − c is a factor?
By the factor theorem, x − c is a factor exactly when the remainder is 0 (p(c) = 0). Then p(x) = (x − c)·q(x), and you can keep factoring q(x).
Why can't I use synthetic division for x² + 1?
Synthetic division only works for linear divisors. For divisors of degree 2 or more you need polynomial long division, which this calculator switches to automatically and shows step by step.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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