How to Use Synthetic Division Calculator
The Synthetic Division Calculator divides a polynomial by a linear factor (x − c) using the efficient synthetic method. Enter the coefficients and the value of c, then read the quotient, remainder, and full step table.
- Enter polynomial coefficients — type all coefficients from highest to lowest degree, separated by spaces or commas. For example, for 2x⁴ − x³ + 0x² + 3x − 7, enter: 2 -1 0 3 -7.
- Enter the divisor root c — type the value of c in the expression (x − c). To divide by (x + 2), enter −2.
- Read the results — the Synthetic Division Calculator shows the synthetic division table, the quotient polynomial, the remainder, and a Remainder Theorem verification P(c) = R.
If the remainder is 0, then c is a root of the polynomial and (x − c) is a factor. You can then use the quotient polynomial to find additional factors.
Formula & Theory - Synthetic Division Calculator
The Synthetic Division Calculator implements the standard synthetic division algorithm:
P(x) ÷ (x − c) = Q(x) + R / (x − c)
Where:
R = P(c) (Remainder Theorem)
Algorithm (row operations):
1. Write the coefficients of P(x): aₙ, aₙ₋₁, ..., a₁, a₀
2. Bring down the leading coefficient aₙ
3. Multiply by c, add to next coefficient → next value in bottom row
4. Repeat until the last value (= R)| Symbol | Meaning |
|---|---|
| P(x) | Dividend polynomial |
| c | Root of the divisor (x − c) |
| Q(x) | Quotient polynomial (degree one less than P) |
| R | Remainder (constant) |
The bottom row of the synthetic division table gives the coefficients of Q(x) (all but the last entry) and the remainder R (the last entry). The Remainder Theorem states that R = P(c), which the Synthetic Division Calculator confirms explicitly.
Assumptions and Limits
- The divisor must be a linear factor of the form (x − c). Higher-degree divisors are not supported by synthetic division.
- Enter a 0 coefficient for any missing term. For example, x⁴ + 1 should be entered as 1 0 0 0 1.
- At least two coefficients are required (representing a polynomial of degree ≥ 1).
Use Cases for Synthetic Division Calculator
The Synthetic Division Calculator is useful across algebra, precalculus, and numerical methods:
- Polynomial factoring — use the Synthetic Division Calculator to test potential rational roots (via the Rational Root Theorem) and factor polynomials step by step.
- Root finding — after finding one root, apply synthetic division to deflate the polynomial and find remaining roots more easily.
- Remainder Theorem problems — evaluate P(c) without substituting directly into the polynomial by reading the remainder from the synthetic division table.
- Algebra homework — verify manual synthetic division work with the full step table provided by the Synthetic Division Calculator.
The table view with color-coded rows makes the Synthetic Division Calculator especially useful for students learning the algorithm for the first time.