Synthetic Division Calculator

Synthetic division: enter a polynomial and a linear divisor (x−c) for compressed steps.

Output: synthetic-division quotient and remainder.

The Synthetic Division Calculator divides a polynomial by a linear factor of the form x − c using the fast synthetic method. Instead of writing out every term, it works only with the coefficients: bring down the first, multiply by c, add to the next, and repeat. The last number is the remainder, and the others are the coefficients of the quotient, whose degree is one less than the dividend. Enter the polynomial's coefficients and the value of c to see each multiply-and-add step, the quotient, the remainder, and the value P(c) by the Remainder Theorem.

How to use the Synthetic Division Calculator

To run synthetic division, follow these 4 steps:

Enter the dividend's coefficients in order, using 0 for any missing power.
Enter the value c from the divisor x − c (use c = −2 for x + 2).
Click Calculate to bring down, multiply by c, and add across.
Read the quotient coefficients and the final remainder.

The divisor must be linear, x − c. For x + 2, set c = −2, because x + 2 = x − (−2).

Synthetic vs. long division

Synthetic division is a shortcut for polynomial long division that only works when the divisor is linear, x − c. It replaces the full divide–multiply–subtract layout with a compact table of coefficients, turning subtraction into addition by using c directly rather than the whole divisor. It is closely tied to the Remainder Theorem: the remainder equals P(c), the value of the polynomial at c. If that remainder is 0, then x − c is a factor of the polynomial, which is the Factor Theorem. The Synthetic Division Calculator makes this quick evaluation and factoring test easy.

Division shares a total into equal groups. Long division does this digit by digit.

Divisor (32) — the number you divide by. Place it to the left of the bracket.

How synthetic division works

The calculator divides by x − c through five internal actions:

Lists the dividend's coefficients in descending order, with zeros for missing powers.
Brings down the leading coefficient unchanged as the first quotient coefficient.
Multiplies the value just written by c and adds it to the next coefficient.
Repeats multiply-and-add across every coefficient.
Reads all but the last result as the quotient coefficients and the last as the remainder.

The quotient has degree one less than the dividend, and the remainder is a constant equal to P(c).

Lists the dividend's coefficients in descending order, with zeros for missing powers.

Formula and the Remainder Theorem

The Synthetic Division Calculator uses the division identity P(x) = (x − c)·Q(x) + r, where the remainder r is a constant. The Remainder Theorem states r = P(c), and the Factor Theorem adds that r = 0 means x − c is a factor. For P(x) = x³ − 2x² − 4 divided by x − 3, synthetic division gives r = P(3) = 5.

Dividend

Divisor

Quotient

Remainder

100 = 7 × 14 + 2 → Identity holds

Synthetic division example problems

These examples divide by a linear factor using multiply-and-add.

Example 1 — (2x² + 3x − 5) ÷ (x − 1)

Coefficients 2, 3, −5 with c = 1; bring down 2.
2 × 1 = 2, add to 3 → 5; 5 × 1 = 5, add to −5 → 0.
Quotient 2x + 5, remainder 0, so x − 1 is a factor.

Example 2 — Missing term: (x³ − 8) ÷ (x − 2)

Coefficients 1, 0, 0, −8 with c = 2; bring down 1.
1 × 2 = 2 → 2; 2 × 2 = 4 → 4; 4 × 2 = 8 → 0.
Quotient x² + 2x + 4, remainder 0.

Example 3 — Divisor x + 2: (x³ − 2x² − 4) ÷ (x + 2)

Use c = −2; coefficients 1, −2, 0, −4; bring down 1.
1 × −2 = −2 → −4; −4 × −2 = 8 → 8; 8 × −2 = −16 → −20.
Quotient x² − 4x + 8, remainder −20.

Coefficients 2, 3, −5 with c = 1; bring down 2.
2 × 1 = 2, add to 3 → 5; 5 × 1 = 5, add to −5 → 0.
Quotient 2x + 5, remainder 0, so x − 1 is a factor.

Worked synthetic division problems

How do you divide x³ − 2x² − 4 by x − 3?

Quotient x² + x + 3, remainder 5. Coefficients are 1, −2, 0, −4 with c = 3. Bring down 1; 1 × 3 = 3, add to −2 → 1; 1 × 3 = 3, add to 0 → 3; 3 × 3 = 9, add to −4 → 5. The remainder 5 equals P(3), confirming the Remainder Theorem.

x^3+2x^2-5x-6x-2

How does the remainder tell you about factors?

If the synthetic-division remainder is 0, then x − c is a factor of the polynomial. For (2x² + 3x − 5) ÷ (x − 1), the remainder is 0, so x − 1 divides it exactly and x = 1 is a root. This is the Factor Theorem, and it makes synthetic division a fast root-testing tool.

x^2+5x+6x+2

Common synthetic division mistakes

Synthetic division produces 5 frequent errors:

Using c with the wrong sign — for x + 2 you must use c = −2.
Omitting a zero coefficient for a missing power, which shifts every result.
Subtracting instead of adding down each column.
Applying synthetic division to a non-linear divisor, where it does not work.
Reading the last number as a quotient coefficient instead of the remainder.

The Synthetic Division Calculator sets the correct sign for c, inserts placeholders, and separates the quotient coefficients from the remainder.

Using c with the wrong sign — for x + 2 you must use c = −2.

Omitting a zero coefficient for a missing power, which shifts every result.

Subtracting instead of adding down each column.

Applying synthetic division to a non-linear divisor, where it does not work.

Reading the last number as a quotient coefficient instead of the remainder.

Frequently Asked Questions

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by a linear factor x − c using only coefficients. You bring down the leading coefficient, multiply by c, add to the next, and repeat; the last value is the remainder.

When can I use synthetic division?

Only when the divisor is linear, of the form x − c. For any higher-degree divisor you must use full polynomial long division instead.

What value of c do I use for x + 2?

Use c = −2, because x + 2 equals x − (−2). Getting this sign right is the most common source of error.

How does synthetic division relate to the Remainder Theorem?

The remainder from dividing P(x) by x − c equals P(c). So synthetic division is also a fast way to evaluate a polynomial at c.

What does a remainder of 0 mean?

By the Factor Theorem, a remainder of 0 means x − c is a factor of the polynomial and c is a root. This makes synthetic division useful for finding roots.

Why do I add instead of subtract?

Synthetic division folds the subtraction of long division into an addition by using c directly instead of the full divisor. That is why each column is added, not subtracted.

What degree is the quotient?

The quotient has degree one less than the dividend. Dividing a cubic by a linear factor gives a quadratic quotient, for example.

The Remainder Theorem connection

The Synthetic Division Calculator satisfies P(x) = (x − c)·Q(x) + r with r = P(c) by the Remainder Theorem, and r = 0 signals that x − c is a factor. For x³ − 2x² − 4 divided by x − 3, the remainder 5 equals P(3), confirming the identity.