Math calculator

Synthetic Division Calculator

This synthetic division calculator divides a polynomial by a linear factor such as x − 1, x + 3 or 2x − 3. Type the polynomial or its coefficients to get the full synthetic division table, the quotient, the remainder, and every step in between.

Perform Synthetic Division

Enter the dividend as a polynomial (x^3 - 2x + 1) or as coefficients in descending order (1, 0, -2, 1).

Enter the divisor as x - 2, x + 3, 2x - 1, or just the value of c.

How to Use the Synthetic Division Calculator

  1. Type the dividend, such as x^3 - 6x^2 + 11x - 6. Missing powers are filled with zeros for you. If you type coefficients instead, include the zeros yourself.
  2. Type the divisor, such as x - 1 or x + 3. A leading coefficient like 2x - 3 is also allowed.
  3. Press Divide with Steps to see the table, the quotient Q(x), the remainder R, and the result written as P(x) = (divisor)·Q(x) + R.
  4. If the remainder is 0, the divisor is a factor, so you can keep factoring the quotient.

How Synthetic Division Works

To divide P(x) by x − c, write c in the corner and the coefficients of P(x) in a row, highest power first. Then repeat one simple cycle:

  1. Bring down the first coefficient.
  2. Multiply the number you just wrote by c and place it under the next coefficient.
  3. Add the column and write the sum in the bottom row.
  4. Repeat until you reach the last column.

The last number in the bottom row is the remainder. The other numbers are the coefficients of the quotient, which has degree one less than P(x). In symbols: P(x) = (x − c)·Q(x) + R.

Two theorems make this useful. The Remainder Theorem says R = P(c), so synthetic division is also a fast way to evaluate a polynomial. The Factor Theorem says that if R = 0, then x − c is a factor and c is a root. For a divisor ax − b, use c = b/a and then divide the quotient coefficients by a.

Worked Examples

Worked example 1: (x³ − 6x² + 11x − 6) ÷ (x − 1)

Use c = 1 and coefficients 1, −6, 11, −6.

 1 |  1   −6   11   −6
   |       1   −5    6
   +--------------------
      1   −5    6 |  0

Bring down 1. 1·1 = 1, −6 + 1 = −5. −5·1 = −5, 11 − 5 = 6. 6·1 = 6, −6 + 6 = 0. Quotient x² − 5x + 6, remainder 0. Since x² − 5x + 6 = (x − 2)(x − 3), the full factorization is (x − 1)(x − 2)(x − 3).

Worked example 2: (2x³ − 3x² + 4) ÷ (x + 2)

The divisor x + 2 means c = −2. There is no x term, so the coefficients are 2, −3, 0, 4.

−2 |  2   −3    0    4
   |      −4   14  −28
   +--------------------
      2   −7   14 | −24

Quotient 2x² − 7x + 14, remainder −24. Check with the Remainder Theorem: P(−2) = 2(−8) − 3(4) + 4 = −24.

Worked example 3: (x⁴ − 16) ÷ (x − 2)

Use c = 2 and coefficients 1, 0, 0, 0, −16. The bottom row is 1, 2, 4, 8 | 0. Quotient x³ + 2x² + 4x + 8, remainder 0, so x − 2 is a factor of x⁴ − 16.

Worked example 4: (2x² + 3x − 5) ÷ (2x − 3)

The divisor is zero at x = 3/2, so use c = 1.5. Bring down 2; 2·1.5 = 3, 3 + 3 = 6; 6·1.5 = 9, −5 + 9 = 4. The bottom row is 2, 6 | 4. Now divide the quotient coefficients by the leading coefficient 2 to get 1, 3. Quotient x + 3, remainder 4, so 2x² + 3x − 5 = (2x − 3)(x + 3) + 4.

Common Mistakes to Avoid

  • Leaving out zero coefficients. x³ + 5 must be entered as 1, 0, 0, 5, not 1, 5.
  • Using the wrong sign for c. For x − 4 use 4; for x + 4 use −4.
  • Subtracting instead of adding. Long division subtracts, but synthetic division adds each column because the sign change is already built into c.
  • Forgetting the leading coefficient. For 2x − 3, divide the quotient by 2 at the end; the remainder does not change.
  • Misreading the degree. The quotient is one degree lower than the dividend: a cubic divided by x − c gives a quadratic.

Frequently Asked Questions

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form x − c. Instead of writing out full polynomial long division, you work only with the coefficients: bring down, multiply by c, add, and repeat. The last number is the remainder and the others are the coefficients of the quotient.

When can you use synthetic division?

Use it when the divisor is linear (degree 1), such as x − 4, x + 2 or 3x − 1. It does not work directly for divisors like x² + 1 or x² − x + 2; use polynomial long division for those.

What does the remainder tell you?

By the Remainder Theorem, dividing P(x) by x − c leaves a remainder equal to P(c). If the remainder is 0, the Factor Theorem says x − c is a factor of P(x) and c is a root. That is why synthetic division is a quick way to test possible roots.

What do I do if a power of x is missing?

Insert 0 as a placeholder for every missing power. For x³ + 5, the coefficients are 1, 0, 0, 5. Skipping the zeros shifts every column and gives a wrong quotient and remainder. The calculator adds these zeros automatically when you type the polynomial.

How do you divide by x + 3 with synthetic division?

Rewrite the divisor as x − (−3), so c = −3. In general, use the number that makes the divisor equal to zero. Dividing by x + 3 means you multiply by −3 at each step, not by +3.

Can synthetic division handle a divisor like 2x − 3?

Yes. Use c = 3/2 (the value that makes 2x − 3 zero), run synthetic division as usual, then divide each quotient coefficient by the leading coefficient 2. The remainder stays the same. For example, (2x² + 3x − 5) ÷ (2x − 3) gives x + 3 with remainder 4.

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How synthetic division works

Formula

divide P(x) by (x − c) using c

Synthetic division is a compact method for dividing a polynomial by a linear factor and finding the remainder.

Worked example

For x² + 3x + 2 divided by x + 1, use c = −1; the quotient is x + 2 with remainder 0.

Common mistake

Use the opposite sign from the divisor and include zero coefficients for missing powers.

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