Polynomial Synthetic Division Calculator

Q: When can I use synthetic division?

Use synthetic division when the divisor is linear and monic, x − c (or x + k which is x − (−k)).

Q: How do I find c for x + 4?

Use c = −4 and plug it into the synthetic division table.

Q: How do I check my result?

Multiply divisor × quotient and add the remainder, and confirm the remainder equals f(c) (Remainder Theorem).

Polynomial Synthetic Division Calculator – Fast Step-by-Step (2025)

Looking for a Polynomial Synthetic Division Calculator? This page is a fast step-by-step guide. You’ll learn exactly when to use synthetic division (and when not to), work through examples, and see how to check your answer. Updated for 2025.

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What Is Synthetic Division?

Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form x − c. It’s quicker than long division because you only track coefficients. The remainder equals f(c) (Remainder Theorem).

Use it when: The divisor is exactly x − c (or x + k which is x − (−k)). Otherwise, use long division.

How to Do Synthetic Division (Step by Step)

Write the coefficients of the dividend in descending powers. Insert zeros for missing powers.
Use c from the divisor x − c. (If the divisor is x + k, then c = −k.)
Bring down the first coefficient.
Multiply that number by c, write the product under the next coefficient, then add.
Repeat multiply–add across the row. The last number is the remainder.
The remaining numbers are the quotient coefficients (one degree lower than the original polynomial).

Pro Tip: Keep columns aligned and watch signs—most mistakes are sign slips.

Worked Example #1 — 2x^3 − 3x^2 + 4x − 5 ÷ (x − 2)

Coefficients: [2, −3, 4, −5], with c = 2.

Bring down 2
2×2 = 4 → add to −3 → 1
1×2 = 2 → add to 4 → 6
6×2 = 12 → add to −5 → 7 (remainder)

Quotient: 2x^2 + x + 6, Remainder: 7

Final Answer: 2x^2 + x + 6 + 7/(x − 2)

Check: Evaluate the original polynomial at x = 2; you get 7 (the remainder).

Worked Example #2 — x^4 − 6x^2 + 9 ÷ (x + 3)

Rewrite coefficients with zeros: [1, 0, −6, 0, 9], and since the divisor is x + 3, use c = −3.

Bring down 1
1×(−3)=−3 → add to 0 → −3
(−3)×(−3)=9 → add to −6 → 3
3×(−3)=−9 → add to 0 → −9
(−9)×(−3)=27 → add to 9 → 36 (remainder)

Quotient: x^3 − 3x^2 + 3x − 9, Remainder: 36

Quick Tip: Remainder Theorem says f(−3)=36, which matches the last box.

What If the Divisor Isn’t x − c?

Divisor ax + b: Rewrite as a(x − (−b/a)). You can use synthetic with c = −b/a but arithmetic may involve fractions. If that’s messy, use long division.
Higher-degree divisor: Synthetic division doesn’t apply. Use long division.

Rule of thumb: Synthetic division is for linear, monic divisors (x − c). Otherwise, use long division.

Common Mistakes (and Fixes)

Forgetting missing powers: Insert coefficients of 0 to keep alignment (e.g., use x^3 + 0x^2 − 4x + 4).
Using the wrong c: For x + k, use c = −k, not k.
Sign slips on the add step: Remember the row operation is always “multiply by c, then add.”
Forgetting the degree drop: Quotient is one degree lower than the dividend.

Check every time: Verify with Dividend = Divisor × Quotient + Remainder.

Helpful Links

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FAQ: Polynomial Synthetic Division Calculator

When can I use synthetic division?

When the divisor is linear and monic, i.e., x − c (or x + k, which is x − (−k)).

How do I find c for x + 4?

Use c = −4. Plug that into the synthetic division table.

Do I need to include missing terms?

Yes. Insert zero coefficients for missing powers so columns line up.

What if the divisor is 2x − 5?

You can use c = 5/2 (fractions may appear), or switch to long division if you want to avoid fractions.

How do I check my result?

Multiply divisor × quotient and add the remainder. Also confirm the remainder equals f(c).