Dividing Polynomials Synthetic Calculator
Polynomial Long & Synthetic Division works out Quotient and Remainder from Input style, Dividend (the polynomial being divided) and Divisor. The default figures on this page give answer of x^2 + 3 + (2)/(x - 2); change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
7 steps
- Divide x^3 - 2x^2 + 3x - 4 by x - 2.
- The divisor is linear, so use synthetic division with the root 2 (from x - 2 = 0).
- Bring down the leading coefficient, multiply by 2, add to the next coefficient, and repeat.
- Bottom row: 1, 0, 3, 2 - every entry but the last is a quotient coefficient, and the last entry is the remainder.
- Quotient x^2 + 3, remainder 2.
- Division algorithm: x^3 - 2x^2 + 3x - 4 = (x - 2)(x^2 + 3) + (2).
- Remainder Theorem: dividing by a linear factor leaves a constant remainder, so evaluating at the root gives it directly - p(2) = 2.
Quotient coefficients (highest power first)table
Remainder coefficients (highest power first)table
Synthetic division tableautable
| column | dividendCoefficient | carried | total |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | -2 | 2 | 0 |
| 3 | 3 | 0 | 3 |
| 4 | -4 | 6 | 2 |
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
What the Polynomial Long & Synthetic Division calculates
Divides one polynomial by another, returning the quotient, remainder, the mixed form q(x) + r(x)/d(x), the long-division working and the synthetic-division tableau for linear divisors.
How to use this Polynomial Long & Synthetic Division
Enter Input style, Dividend (the polynomial being divided), Divisor and Dividend coefficients. Quotient updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Quotient, Remainder and Remainder (numeric, constant remainders only).
The method follows OpenStax, College Algebra 2e, Section 5.4: Dividing Polynomials (long division, synthetic division, the Division Algorithm) and OpenStax, College Algebra 2e, Section 5.5: the Remainder Theorem and Factor Theorem.
(x^3 - 2x^2 + 3x - 4) / (x - 2) example
With Dividend (the polynomial being divided): x^3 - 2x^2 + 3x - 4, Divisor: x - 2, the calculator returns Quotient: x^2 + 3, Remainder: 2, Remainder (numeric, constant remainders only): 2. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Quotient, Remainder and Remainder (numeric, constant remainders only). The inputs you can vary are Input style, Dividend (the polynomial being divided), Divisor, Dividend coefficients, Divisor coefficients and Coefficient order.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
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Frequently asked questions
How to calculate synthetic division?
Divides one polynomial by another, returning the quotient, remainder, the mixed form q(x) + r(x)/d(x), the long-division working and the synthetic-division tableau for linear divisors. Open “How this is worked out” beside the result to see each step for your own figures.
Can a TI-84 solve polynomials?
Divides one polynomial by another, returning the quotient, remainder, the mixed form q(x) + r(x)/d(x), the long-division working and the synthetic-division tableau for linear divisors. Put your own figures in above and the working updates beside the result. For scale, the default figures on this page give x^2 + 3 + (2)/(x - 2).
Can you explain synthetic division for dummies?
It needs Input style, Dividend (the polynomial being divided), Divisor, Dividend coefficients, Divisor coefficients and Coefficient order. Nothing else is required, and the figures stay in your browser.
How do I know if I can use synthetic division?
The result gives Quotient, Remainder, Remainder (numeric, constant remainders only), Answer, Division algorithm check and Divides exactly, each labelled with its unit so the figure can be read on its own.
What information do I need for the Polynomial Long & Synthetic Division?
You need Input style, Dividend (the polynomial being divided), Divisor, Dividend coefficients, Divisor coefficients, Coefficient order, Method and Variable. The page opens with default figures that give answer of x^2 + 3 + (2)/(x - 2), so you can see the scale before entering your own.
What does this Polynomial Long & Synthetic Division show?
It shows Quotient, Remainder, Remainder (numeric, constant remainders only), Answer, Division algorithm check, Divides exactly, Degree of the quotient and Degree of the remainder, with the working and any warnings for your entries.