Synthetic Division Calculator
Here is what synthetic division looks like when it is finished. Divide 2x³ − 5x² + 3x − 7 by x − 2 — the division this synthetic division calculator is built for — and this is the whole of the work:
2 │ 2 −5 3 −7
│ 4 −2 2
──────────────
2 −1 1 −5
Four numbers along the top, three products, four sums. The quotient is 2x² − x + 1 and the remainder is −5. No x anywhere, no subtraction signs, no rewriting the same terms four times. The synthetic division calculator above builds that tableau column by column for whatever you type. Most people looking for a calculator for synthetic division want the steps rather than the answer, so every column is shown as it is built — and then, the part no other tool does, the same division is worked again as long division underneath, so you can see they are the same thing.
Reading That Tableau, One Column at a Time
Take the same division apart. The 2 on the left is not the divisor. It is the root — the value that makes x − 2 equal zero. Set the divisor to zero, solve, and that is the number you write outside.
The top row is the coefficients of the dividend: 2, −5, 3, −7. Then:
Bring the 2 straight down.
2 × 2 = 4, written above the −5. Add: −5 + 4 = −1.
−1 × 2 = −2, written above the 3. Add: 3 − 2 = 1.
1 × 2 = 2, written above the −7. Add: −7 + 2 = −5.
Bottom row: 2, −1, 1, −5. The first three are the quotient coefficients — one degree lower than you started, so 2x² − x + 1. The last one, −5, is the remainder.
The Sign Flip That Costs Marks
You divide by x − 2 and run the tableau with +2. You divide by x + 3 and run it with −3. The number you write outside is the root of the divisor, not the number printed inside the bracket.
It reads as a trick until you see why: synthetic division is really evaluating the polynomial at that root, and the root of x + 3 is −3. If your remainder comes out with the wrong sign, this is almost always the reason.
Missing Terms Are Not Optional
x³ − 1 has no x² term and no x term. Its coefficients are 1, 0, 0, −1 — four slots, not two.
Every power from the highest down to the constant needs a column. Skip a gap and every column after it shifts left, so the products land on the wrong numbers and everything from that point on is wrong. It is the single commonest way a synthetic division goes bad, and it is silent — the arithmetic still works, it just answers a different question. The calculator fills the zeros in for you and shows them in the tableau so you can see where they went.
Dividing by 2x − 1 and Other Non-Monic Divisors
Most tools stop here. Two of the biggest state it outright: the divisor must be x − c. But exam papers use 2x − 1 and 3x + 2 all the time, so it is worth knowing that the method does handle them.
Run the tableau with the root as normal — for 2x − 1 that is ½ — and then divide the quotient row by the leading coefficient, 2. The remainder is not divided.
Try 6x³ + 5x² − 7 divided by 2x − 1 in the calculator. The tableau gives 6, 8, 4 and a remainder of −5; halving the first three gives the quotient 3x² + 4x + 2. Multiply back and you get the original polynomial. Forget to halve, and your quotient is twice what it should be — a mistake that survives every check except multiplying back out.
Synthetic Division vs Long Division
These are not two methods. They are one method written two ways, which is why the long and synthetic division of your polynomial are both shown here for the same problem, side by side. The long division shown below is the linear-divisor case; for a divisor of degree 2 or more there is a fuller layout, and that is a job for a dedicated polynomial long division calculator.
| Long division | Synthetic division | |
|---|---|---|
| What you write | terms, with the variable | coefficients only |
| The operation | multiply and subtract | multiply and add |
| Divisor allowed | any degree | degree 1 only |
| Speed | slow, hard to slip | fast, easy to slip |
| Answer | identical | identical |
The subtraction is where the sign flip comes from. Long division subtracts each row; synthetic division negates the divisor's constant once, at the start, and then only ever adds. Same arithmetic, one fewer thing to get wrong — and one more thing to forget.
When Synthetic Division Will Not Work
The divisor has to be degree 1. Divide x³ − 1 by x² + x + 1 and there is no tableau to build — there is no single root to put outside the bracket.
Most calculators either reject the input or fail without explanation. This one switches to long division, tells you why, and gives you the answer anyway: the quotient is x − 1, with no remainder. If you need a full division of any degree rather than this one method, the polynomial calculator handles addition, subtraction, multiplication and division together.
How to Use This Synthetic Division Calculator
This is a polynomial synthetic division calculator, so both boxes take polynomials rather than bare numbers. To divide using synthetic division, put the polynomial in the first box and the divisor in the second, then press =. Missing terms are filled in for you, terms can be in any order, and fractions and decimals both work. Tap either box to aim the keypad at it; the key icon beside a box switches that one to your own keyboard.
The blue band gives the quotient and remainder, with the tableau on the white plate below it — the quotient coefficients in blue, the remainder in red. Under that, the working: finding the root, laying out the coefficients, the bring-down-multiply-add loop, and then the same division done the long way for comparison. The last step multiplies everything back out to check it.
Both methods are actually run, every time, and their answers compared before anything appears on screen. If they ever disagreed, the calculator would show nothing rather than pick one.
What the Remainder Is Telling You
Divide by x − 2 and get a remainder of −5, and you have also learned that the polynomial equals −5 when x = 2. That is the remainder theorem, and it makes synthetic division a fast way to evaluate a polynomial — sometimes called synthetic substitution.
A remainder of zero means the divisor is a factor, which is how synthetic division earns its keep in factoring. If that is the question you actually came with — the value of the remainder, or whether something is a factor — the remainder theorem calculator is built around that question specifically. This page is about the division itself. And once you have a factor, the factoring polynomials calculator takes the polynomial the rest of the way.
Worked Examples
| Division | Quotient | Remainder | Watch for |
|---|---|---|---|
| 2x³ − 5x² + 3x − 7 ÷ x − 2 | 2x² − x + 1 | −5 | the plain case |
| x⁴ − 5x³ + 7x² − 34x − 1 ÷ x − 5 | x³ + 7x + 1 | 4 | quotient has no x² term |
| x³ − 1 ÷ x − 1 | x² + x + 1 | 0 | two missing terms |
| 6x³ + 5x² − 7 ÷ 2x − 1 | 3x² + 4x + 2 | −5 | halve the quotient, not the remainder |
| x² − 4 ÷ x + 2 | x − 2 | 0 | run it with −2, not +2 |
| x³ − 1 ÷ x² + x + 1 | x − 1 | 0 | long division only |
Frequently Asked Questions
How do you do synthetic division step by step?
Why does the sign flip in synthetic division?
What do you do about missing terms in synthetic division?
Can you use synthetic division with 2x − 1?
When can you not use synthetic division?
What is the difference between synthetic division and long division?
Is this synthetic division calculator free?
Synthetic division is worth the ten minutes it takes to learn properly. Get the root sign right, put a zero in every gap, and remember to divide the quotient row when the divisor is not monic — those three habits cover almost every mark lost on it. Use the calculator above to watch the tableau build while you are still forming them.