Math July 13, 2026 · 8 Min Read

Synthetic Division Calculator – Guide & Formulas

Perform synthetic division on any polynomial. Enter coefficients and divisor to see the complete synthetic division table with quotient and remainder.

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Put these formulas into practice with our instant, step-by-step Synthetic Division Calculator.

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Use our free **Synthetic Division Calculator** to **perform polynomial division** with step-by-step results. This **synthetic division solver** helps you **divide polynomials by binomials** and find **quotients and remainders** accurately. Whether you need to **verify roots** or **factor polynomials**, this tool provides complete breakdowns. Features include **synthetic division table**, **remainder theorem verification**, and **coefficient tracking**. 100% free — no signup required!

Key Takeaway

Use the free Synthetic Division Calculator to perform synthetic division on any polynomial. enter coefficients and divisor to see the complete synthetic division table with quotient and remainder. Get instant results with step-by-step explanations.

How to Use the Synthetic Division Calculator

  1. Step 1: Enter the **polynomial coefficients** from highest degree to constant term (e.g., 1, -6, 11, -6 for x³ - 6x² + 11x - 6) into the synthetic division calculator.
  2. Step 2: Enter the **divisor value r** (for dividing by x - r). If dividing by x - 2, enter r = 2.
  3. Step 3: Review the **synthetic division table** showing each multiplication and addition step.
  4. Step 4: Follow the **synthetic division steps** as the calculator brings down coefficients and computes products.
  5. Step 5: Check the **quotient coefficients** and **remainder** displayed at the bottom row of the table.
  6. Step 6: Verify the result using the **remainder theorem**: confirm that the remainder equals P(r).
  7. Step 7: Use the step-by-step breakdown to understand how the **polynomial division shortcut** works.

The Formula

Synthetic division divides polynomial P(x) by (x - r) to get quotient Q(x) and remainder R. The bottom row of the table gives the quotient coefficients (all but last value) and remainder (last value). By the Remainder Theorem, R = P(r).

Variable Definitions

  • r: The divisor value (dividing by x - r)
  • Q(x): The quotient polynomial (degree one less than P(x))
  • R: The remainder (a constant, equals P(r) by Remainder Theorem)
  • Bottom row: Contains quotient coefficients followed by the remainder
  • Coefficients: Numbers in front of each power of x, including 0 for missing terms

Dividing x³ - 6x² + 11x - 6 by x - 2

Perform synthetic division with r = 2.

  1. Step 1: Write coefficients: 1, -6, 11, -6. Divisor r = 2.
  2. Step 2: Bring down the leading coefficient: 1.
  3. Step 3: Multiply 1 × 2 = 2, add to -6: -6 + 2 = -4.
  4. Step 4: Multiply -4 × 2 = -8, add to 11: 11 + (-8) = 3.
  5. Step 5: Multiply 3 × 2 = 6, add to -6: -6 + 6 = 0. Quotient: x² - 4x + 3, Remainder: 0.

Frequently Asked Questions

What is synthetic division?

Synthetic division is a shortcut method for dividing a polynomial by a linear factor (x - r). It uses only the coefficients and is faster than polynomial long division. The result gives the quotient and remainder.

When can I use synthetic division?

Only when dividing by a linear factor of the form (x - r). For higher-degree divisors like (x² + 1), use polynomial long division instead.

How do I handle missing terms?

Insert 0 as the coefficient for any missing power. For example, x³ + 1 has coefficients 1, 0, 0, 1 (missing x² and x terms).

What does a remainder of 0 mean?

A remainder of 0 means (x - r) is a factor of the polynomial. By the Factor Theorem, P(r) = 0, so r is a root of the polynomial.

How does this relate to the Remainder Theorem?

The Remainder Theorem states that when dividing P(x) by (x - r), the remainder equals P(r). So synthetic division gives you both the quotient and a way to evaluate the polynomial at x = r.

What if r is negative?

If dividing by (x + 3), then r = -3. Use r = -3 in the synthetic division. The sign of r is always opposite to the constant in (x - r).

Can I use synthetic division for higher-degree divisors?

No. Synthetic division only works for linear divisors (x - r). For quadratic or higher-degree divisors, use polynomial long division or other methods.

How do I interpret the bottom row?

The bottom row contains the coefficients of the quotient (all values except the last) and the remainder (last value). If dividing a degree-n polynomial, the quotient has degree n-1.

How do I use synthetic division to find polynomial roots?

Try values from the Rational Root Theorem. If synthetic division gives a remainder of 0, that value is a root. Repeat with the quotient polynomial to find additional roots.

What is the difference between synthetic and long division?

Synthetic division is a shortcut that only works for linear divisors (x - r). Long division works for any divisor degree but is more tedious. Synthetic division is faster and less error-prone for linear factors.