Quadratic Formula Calculator — Solve ax² + bx + c = 0
Solve any quadratic equation with the quadratic formula. Find both roots, discriminant, vertex, axis of symmetry, and factored form with step-by-step solutions.
Synthetic Division Calculator
Divide polynomials using synthetic division. Enter coefficients and divisor to get quotient, remainder, step-by-step layout, factor theorem test, and rational root candidates.
Synthetic Division Calculator
Dividend⧉
x^3 −6x^2 +11x −6
Divisor⧉
(x − 2.00)
Quotient⧉
x^2 −4x +3
Degree 2 polynomial
Remainder⧉
0.0000
(x − c) is a factor!
f(c) by Remainder Theorem⧉
0.0000
f(2) = 0.0000
Factor Theorem⧉
✅ (x − c) IS a factor
x^3 −6x^2 +11x −6 = (x − 2) · x^2 −4x +3
Synthetic Division Steps
| Step | Col 1 | Col 2 | Col 3 | Col 4 |
|---|---|---|---|---|
| Coefficients | 1.0000 | -6.0000 | 11.0000 | -6.0000 |
| × 2.00 | 2.0000 | -8.0000 | 6.0000 | |
| Result | 1.0000 | -4.0000 | 3.0000 | 0.0000 |
Coefficient Magnitude
x^3 coeff: 1.001.00
x^2 coeff: -6.006.00
x^1 coeff: 11.0011.00
x^0 coeff: -6.006.00
Result Breakdown
| Term | Coefficient | Degree | Role |
|---|---|---|---|
| 1.00x^2 | 1.0000 | 2 | Quotient term |
| -4.00x | -4.0000 | 1 | Quotient term |
| 3.00 | 3.0000 | 0 | Quotient term |
| 0.0000 | 0.0000 | — | Remainder |
Verification
x^3 −6x^2 +11x −6 = (x − 2.00) · (x^2 −4x +3)
Possible Rational Roots (Rational Root Theorem)
±p/q: 1/1, -1/1, 2/1, -2/1, 3/1, -3/1, 6/1, -6/1
Planning notes, formulas, and examples
About the Synthetic Division Calculator
Synthetic division is a streamlined shortcut for dividing a polynomial by a linear binomial of the form (x − c). Instead of performing lengthy polynomial long division, synthetic division uses only the coefficients of the dividend and the value c, dramatically reducing the amount of writing and arithmetic involved. The process produces both the quotient polynomial and the remainder in a single compact table.
Beyond simple division, synthetic division is the practical engine behind the Remainder Theorem and the Factor Theorem. The Remainder Theorem states that when a polynomial f(x) is divided by (x − c), the remainder equals f(c). The Factor Theorem extends this: if f(c) = 0, then (x − c) is an exact factor of f(x). Together, these theorems let you test possible roots quickly. Combined with the Rational Root Theorem—which lists all candidates ±p/q where p divides the constant term and q divides the leading coefficient—synthetic division becomes the fastest way to factor higher-degree polynomials by hand.
This calculator accepts any polynomial up to degree 20+, shows the full synthetic-division layout step by step, reports quotient and remainder, tests the Factor Theorem, and lists possible rational roots. Use the presets to explore classic textbook examples or enter your own coefficients.
When This Page Helps
Synthetic Division Calculator helps you solve synthetic division problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficients (comma-separated, highest degree first), Divisor c in (x − c), Decimal places once and immediately inspect Dividend, Divisor, Quotient to validate your work.
How to Use the Inputs
- Enter Coefficients (comma-separated, highest degree first) and Divisor c in (x − c) in the input fields.
- Select the mode, method, or precision options that match your synthetic division problem.
- Read Dividend first, then use Divisor to confirm your setup is correct.
- Try a preset such as "x³−6x²+11x−6 ÷ (x−1)" to test a known case quickly.
Formula used
Given P(x) = aₙxⁿ + … + a₁x + a₀ and divisor (x − c), synthetic division produces Q(x) and remainder R such that P(x) = (x − c)·Q(x) + R. The remainder R also equals P(c) (Remainder Theorem).Example Calculation
Result: Dividend shown by the calculator
Using the preset "x³−6x²+11x−6 ÷ (x−1)", the calculator evaluates the synthetic division setup, applies the selected algebra rules, and reports Dividend with supporting checks so you can verify each transformation.
Tips & Best Practices
- Always include zero coefficients for missing powers (e.g., x³ − 1 → 1, 0, 0, −1).
- If dividing by (x + a), use c = −a in the input.
- A remainder of 0 confirms (x − c) is a factor—use the quotient to continue factoring.
- Combine with the Rational Root Theorem to systematically test every possible rational root.
How This Synthetic Division Calculator Works
This calculator takes Coefficients (comma-separated, highest degree first), Divisor c in (x − c), Decimal places and applies the relevant synthetic division relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use Dividend, Divisor, Quotient, Remainder to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
Synthetic division is a simplified method for dividing a polynomial by a linear binomial (x − c). It uses only coefficients and basic arithmetic instead of the full long-division setup.
Yes. Rewrite (x + 3) as (x − (−3)) and use c = −3 in the calculator.
A remainder of 0 means (x − c) is an exact factor of the polynomial, and f(c) = 0 by the Factor Theorem.
Insert 0 for any missing degree. For example, x⁴ − 1 should be entered as 1, 0, 0, 0, −1.
Yes. You can enter any real number for c, including decimals and fractions (as decimals). The arithmetic works the same way.
The Rational Root Theorem states that any rational root p/q of a polynomial with integer coefficients must have p dividing the constant term and q dividing the leading coefficient. It gives a finite list of candidates to test via synthetic division.
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