Synthetic division is a compact shortcut for dividing a polynomial by a linear factor x − c. Instead of writing out every power of x, you work only with the coefficients: bring one down, multiply by c, add, and repeat. This calculator builds the table, reads off the quotient and remainder, and confirms the result with the remainder theorem.
How to use the synthetic division calculator
- Enter the dividend, either as a polynomial (
2x^3 - 5x^2 + 3x + 7) or as its coefficients from the highest power down (2 -5 3 7). - Enter the divisor:
x - 2,x + 3,2x - 1, or just the number c for x − c. - Read the quotient and remainder on the tape. The table below shows the three rows of the method, with the remainder highlighted at the end.
How synthetic division works
Write c in a box and the coefficients in a row beside it. Then:
- Bring down the first coefficient unchanged.
- Multiply that number by c and write the product under the next coefficient.
- Add the column and write the sum in the bottom row.
- Repeat multiply-and-add until you reach the last column.
The bottom row, apart from its last entry, holds the coefficients of the quotient, which has a degree one less than the dividend. The last entry is the remainder. The result satisfies
Worked example
Divide 2x3 − 5x2 + 3x + 7 by x − 2, so c = 2.
Coefficients: 2, −5, 3, 7.
Bring down 2.
2 × 2 = 4; −5 + 4 = −1.
−1 × 2 = −2; 3 + (−2) = 1.
1 × 2 = 2; 7 + 2 = 9 (the remainder).
Answer: quotient 2x2 − x + 1, remainder 9.
Check with the remainder theorem: P(2) = 2(8) − 5(4) + 3(2) + 7 = 16 − 20 + 6 + 7 = 9. ✓
Finding roots and factoring
Synthetic division is the standard tool for testing candidate roots. The rational root theorem says any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient. Test each candidate; a remainder of 0 confirms a root.
Take x3 − 6x2 + 11x − 6. Testing c = 3 gives the bottom row 1, −3, 2, 0, so 3 is a root and the polynomial factors as (x − 3)(x2 − 3x + 2) = (x − 1)(x − 2)(x − 3). Once a cubic is reduced to a quadratic, the quadratic formula calculator finishes the job; the cubic equation calculator solves cubics directly.
Dividing by ax + b
When the divisor’s leading coefficient is not 1, factor it out: 2x − 1 = 2(x − ½). Run synthetic division with c = ½, then divide every quotient coefficient by 2. The remainder stays the same. For 2x3 + x2 − 13x + 6 ÷ (2x − 1), the bottom row is 2, 2, −12, 0, so the quotient is x2 + x − 6 with no remainder.
Common mistakes
- Using the wrong sign for c. For x + 4 the box holds −4, because x + 4 = x − (−4).
- Leaving out zero coefficients. x4 − 16 must be entered as 1, 0, 0, 0, −16.
- Subtracting instead of adding. Unlike long division, synthetic division always adds the column.
For divisors of degree two or more, switch to the polynomial long division calculator.
Frequently asked questions
What divisors work with synthetic division?
Only linear ones: x − c, x + c, or ax + b. For x + 3 the number in the box is −3. For a divisor such as 2x − 1, divide by x − 1/2 and then divide the quotient's coefficients by 2.
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 number.
How do I know if c is a root?
If the remainder is 0, then P(c) = 0, so c is a root and x − c is a factor (the factor theorem). The quotient is the remaining factor.
Can I type just the coefficients?
Yes. Enter them from the highest power down, such as 1 −6 11 −6, and include 0 for any missing power. You can also type the polynomial itself, in which case missing powers are filled in automatically.