What this calculator does
Synthetic division is a shorthand for dividing a polynomial by a linear factor. It strips away the variable entirely and works only with coefficients, which makes it far quicker than long division for this specific case.
The remainder is more useful than it looks. By the remainder theorem it equals the polynomial evaluated at r, so a remainder of zero means x minus r is a factor and r is a root.
The formula
The leading coefficient is brought down. Each subsequent value is the previous result multiplied by r and added to the next coefficient. The final value is the remainder and the rest are the quotient coefficients.
| Term | Meaning |
|---|---|
| Remainder theorem | The remainder on dividing by x minus r equals the polynomial evaluated at r. |
| Factor theorem | A remainder of zero means x minus r divides the polynomial exactly. |
| Quotient | The polynomial result of the division, one degree lower than the original. |
The inputs explained
| Field | What to enter |
|---|---|
| Coefficients, highest degree first | Coefficients from the highest degree down, comma separated. Include zeros for missing terms. |
| Divide by (x − r): value of r | The value of r in the divisor x minus r. This is the candidate root being tested. |
When to use it
Testing a candidate root
A remainder of zero confirms the value is a root.
Factorising a polynomial
Each root found reduces the degree, making the rest easier.
Evaluating a polynomial quickly
The remainder theorem makes this faster than substituting directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Which values are roots?
Four candidate divisors tested against the same polynomial.
| Value of r | Remainder | Quotient |
|---|---|---|
| r = 1 | 0 | 1.00x^2 − 5.00x + 6.00 |
| r = 2 | 0 | 1.00x^2 − 4.00x + 3.00 |
| r = 3 | 0 | 1.00x^2 − 3.00x + 2.00 |
| r = 4 | 6.000 | 1.00x^2 − 2.00x + 3.00 |
Questions
Why does the remainder equal the polynomial at r?
Because dividing gives P(x) = (x − r)Q(x) + R. Substituting x = r makes the first term vanish, leaving P(r) = R. That identity is the remainder theorem, and it makes synthetic division a fast evaluation method.
What if a degree is missing?
Enter a zero coefficient for it. A polynomial like x³ + 1 needs coefficients 1, 0, 0, 1, and omitting the zeros shifts every subsequent term and gives a wrong answer.
Does this work for non-linear divisors?
Not in this form. Synthetic division is specific to divisors of the form x minus r. Dividing by a quadratic or higher needs polynomial long division, though extended variants of synthetic division exist.
How do I find candidate roots?
The rational root theorem narrows the list: any rational root must be a factor of the constant term divided by a factor of the leading coefficient. Testing those candidates by synthetic division is the standard approach.
For solving cubics directly, see the cubic equation calculator. For quadratics, see the quadratic calculator.