Synthetic Division Remainder Calculator

Master polynomial division with an intuitive, professional calculator interface for experts. Input coefficients, select divisor form, and see synthetic tables with clear operations. Get quotient terms, remainder value, and verification via Remainder Theorem for accuracy. Export CSV and PDF for records, audits, sharing easily.

Input
Separate by comma or space. Example: 2 -3 0 5 -7
Please enter at least one numeric coefficient.
Divisor
Example Data

Try these examples by copying the coefficients and divisor values:

Polynomial (coefficients) Divisor Expected remainder
2, -3, 0, 5, -7 x - 3 P(3)
1, 0, -4, 4 x - 2 P(2)
3, -1, -7, 10 2x - 6 P(3) (since r = b/a = 3)
Results
Polynomial: P(x) = 2 x^4 - 3 x^3 + 5 x - 7
Divisor: x - (3)
Using r: 3
Remainder (Remainder Theorem):
89
Computed via Horner's method / synthetic division.
Quotient (for divisor x - r):
Q(x) = 2 x^3 + 3 x^2 + 9 x + 32
For a·x - b, quotient scaling differs; remainder is valid.
Synthetic Division Table
r 2 -3 0 5 -7
3 6 9 27 96
Sum 2 3 9 32 89
Formula Used

Remainder Theorem: When dividing a polynomial P(x) by (x - r), the remainder equals P(r).

Horner’s Method / Synthetic Division: For coefficients [a0, a1, ..., an] and root r:

b0 = a0;   bk = ak + r · bk-1 for k = 1..n

The remainder is bn and the quotient coefficients are [b0, ..., bn-1]. For divisors of the form a·x - b, use r = b / a for the remainder.

How to Use
  1. Enter polynomial coefficients from highest degree to constant term.
  2. Select the divisor form and fill in its parameter(s).
  3. Click Calculate to compute remainder and quotient.
  4. Toggle Show steps to view the synthetic division table.
  5. Use Download CSV or Download PDF to export results.
FAQs

Enter coefficients separated by commas or spaces, starting from the highest degree to the constant term.

Yes. The remainder is computed using r = b/a. The displayed quotient is valid for x − r; a·x − b requires scaling for the quotient, but the remainder remains correct.

By the Remainder Theorem, evaluating the polynomial at r yields the remainder when dividing by x − r, which Horner’s method computes efficiently.

You can set decimal precision (0–10). Internally, computations use double precision and then format to your chosen precision.

Yes, for divisors of the form x − r. For a·x − b, the remainder is still correct; quotient coefficients need scaling not shown here.
Quick Reference Examples
Polynomial r Remainder Notes
P(x) = 2x^4 - 3x^3 + 5x - 7 3 89 From coefficients [2,-3,0,5,-7].
P(x) = x^3 - 4x + 4 2 4 From coefficients [1,0,-4,4].
P(x) = 3x^3 - x^2 - 7x + 10 3 61 Divisor 2x - 6 gives r = 3.
P(x) = x^3 - 6x^2 + 11x - 6 1 0 Root at 1 ⇒ remainder zero.
Operation Count by Degree (Horner’s Method)

For degree n: multiplications = n, additions = n. Remainder equals P(r).

Degree n Multiplications Additions
111
222
333
444
555
666
Divisor Form Reference
Divisor Use r = Remainder Valid Quotient Displayed Notes
x - r r Yes Yes Standard synthetic division.
a·x - b b / a Yes Not directly Quotient needs scaling by a across steps.
x + c -c Yes Yes Special case of x - r.

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