Synthetic Division Calculator: How to Divide a Polynomial by x − c
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A Synthetic Division Calculator divides a polynomial by a linear factor of the form x − c in a single compact table, returning the quotient and remainder without the bulky bookkeeping of long division. Feed it 2x³ − 5x² + 3x − 7 and a divisor of x − 3, and it hands back 2x² + x + 6 with a remainder of 11 before you finish reading this sentence. This guide breaks down exactly how that table works, why the last number in it is secretly the value of the polynomial, and where the shortcut quietly fails.
What Synthetic Division Actually Does
Synthetic division is a streamlined version of polynomial long division that only works for one specific kind of divisor: a linear binomial x − c. Instead of writing out x terms over and over, you strip the polynomial down to its coefficients and run a short loop of multiply-and-add. The payoff is speed. A cubic that takes half a page with long division collapses into three rows of arithmetic.
The trade-off is scope. Because the method assumes the divisor is x minus a single number, it cannot directly divide by something like x² + 1 or 3x − 6 without a tweak. For those, you still want full polynomial long division. Synthetic division is the scalpel; long division is the toolbox.
The Algorithm: Bring Down, Multiply, Add
Every synthetic division follows the same three-beat rhythm. Once you internalize it, the calculator above just becomes a faster pair of hands.
- Set up the row. Write the coefficients of the dividend in order of decreasing power. If a power is missing — say there is no x² term — write a 0 in its place. Skipping that zero is the single most common way the whole table goes wrong.
- Bring down the leading coefficient unchanged into the bottom row.
- Multiply that bottom value by c, write the product under the next coefficient, then add the column. Repeat across the row.
When you reach the last column, the bottom number is the remainder and everything to its left is the quotient, one degree lower than where you started. That is the whole engine.
A Worked Example With a Nonzero Remainder
Take p(x) = 2x³ − 5x² + 3x − 7 divided by x − 3, so c = 3. Lay out the coefficients 2, −5, 3, −7. Bring down the 2. Multiply 2 × 3 = 6 and add to −5 to get 1. Multiply 1 × 3 = 3 and add to 3 to get 6. Multiply 6 × 3 = 18 and add to −7 to get 11.
The bottom row reads 2, 1, 6, 11. Reading it back: the quotient is 2x² + x + 6 and the remainder is 11. So 2x³ − 5x² + 3x − 7 = (x − 3)(2x² + x + 6) + 11. If you want to push the quotient further toward its roots, drop 2x² + x + 6 into the quadratic equation calculator and let it finish the job.
| Step | Operation | Running bottom row |
|---|---|---|
| 1 | Bring down 2 | 2 |
| 2 | 2 × 3 = 6, then −5 + 6 | 2, 1 |
| 3 | 1 × 3 = 3, then 3 + 3 | 2, 1, 6 |
| 4 | 6 × 3 = 18, then −7 + 18 | 2, 1, 6, 11 |
The Remainder Theorem Hiding in the Last Column
Here is the part most students miss. That remainder of 11 is not just leftover arithmetic — it is exactly p(3). Plug 3 into the original polynomial: 2(27) − 5(9) + 3(3) − 7 = 54 − 45 + 9 − 7 = 11. Same number. This is the Remainder Theorem, and it turns synthetic division into the fastest way to evaluate a polynomial at a point.
The Factor Theorem rides shotgun. If the remainder comes out to 0, then x − c divides evenly and c is a root. That is why the calculator flags whether (x − c) is a factor: a zero remainder means you have found a root and can peel it off, then keep factoring the smaller quotient with a factoring polynomials calculator.
Synthetic Division Calculator vs. Polynomial Long Division
Both methods produce the same quotient and remainder, but they earn their keep in different situations. Use this table to pick the right tool before you start writing.
| Factor | Synthetic division | Long division |
|---|---|---|
| Divisor allowed | Only x − c (linear, leading coefficient 1) | Any polynomial divisor |
| Speed on a cubic | 3 rows, ~5 operations | Multiple subtraction blocks |
| Error surface | Low, but sign of c trips people up | Higher, more places to slip |
| Best for | Root testing, repeated evaluation | Quadratic or higher divisors |
For broader symbolic work beyond division — combining, expanding, or simplifying expressions — a general polynomial calculator covers the operations synthetic division does not.
The Trick for Dividing by ax − b
Suppose your divisor is 2x − 1 instead of x − c. You cannot feed 2x − 1 to synthetic division directly, but you can rewrite it as 2(x − ½). Run synthetic division with c = ½, then divide every quotient coefficient by 2 to undo the factor of 2 you pulled out. The remainder is unaffected. It is a small two-step dance, and it lets the shortcut handle divisors that look off-limits at first glance.
Common Mistakes That Wreck the Tableau
- Forgetting zero placeholders. Dividing x⁴ − 16 means coefficients 1, 0, 0, 0, −16 — four zeros for the missing x³, x², and x terms. Omit them and every column after shifts, giving nonsense.
- Using the wrong sign for c. Dividing by x + 5 means c = −5, not 5. The divisor is x − c, so flip the sign of the constant you see.
- Treating the remainder as a coefficient. The last bottom-row number is the remainder, not part of the quotient. The quotient always sits one degree below the dividend.
- Adding instead of multiplying (or vice versa). The rhythm is multiply across the diagonal, add down the column. Swap them and the table quietly produces a believable but wrong answer.
When This Shortcut Is the Right Call
Reach for synthetic division when you are testing candidate roots from the rational root theorem, evaluating a polynomial at several points, or peeling a known linear factor off a high-degree expression before handing the rest to a factor calculator. It shines in Algebra 2, precalculus, and any moment where you need p(c) fast. When the divisor stops being linear, switch back to long division — and for a deeper proof of why the method works, the Wikipedia entry on synthetic division walks through the underlying algebra.
