# How do I do synthetic division?

Divides a polynomial by a linear divisor such as x − 3 or 2x + 1 with synthetic division, in exact fractions: the quotient, the remainder, p(k), and every bring-down, multiply and add step.

- Page: https://www.acalculator.org/math/synthetic-division-calculator
- JSON spec: https://www.acalculator.org/math/synthetic-division-calculator.json
- Version: ae209fa70b1c

## Default answer

Example with the default inputs (Dividend 5x^2 - 3x - 36, Divisor x - 3): (5x^2 - 3x - 36) ÷ (x - 3) = 5x + 12, remainder 0.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| p | Dividend | The polynomial to divide, in one letter. |
| d | Divisor | A divisor of degree 1, such as x - 3, x + 2 or 2x - 1, in the same letter. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| quotient | Quotient | The dividend ÷ the divisor, without the remainder. |
| remainder | Remainder | What is left over: p(k), a number. |
| k | k | The number synthetic division uses: the zero of the divisor. |
| value | Remainder theorem | The dividend’s value at k, which equals the remainder. |
| row | Bottom row | The numbers under the line, the last one the remainder. |
| steps | Steps | Every step of the synthetic division. |

## Method

For a divisor ax + b, k = −b ÷ a. Bring down the first coefficient; then multiply by k and add to the next coefficient, to the end. The last number is the remainder; the others, divided by a, are the quotient.

## Assumptions

- Coefficients are exact fractions: 0.5 is 1/2.
- The divisor has degree 1; the dividend has degree 1 to 30, in the same letter.

## Worked examples

1. p = 5x^2 - 3x - 36, d = x - 3 gives quotient = 5x + 12, remainder = 0, k = 3, row = 5, 12 | 0, value = p(3) = 0. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 3 Using Synthetic Division to Divide a Second-Degree Polynomial: 5x + 12, remainder 0.
2. p = 4x^3 + 10x^2 - 6x - 20, d = x + 2 gives quotient = 4x^2 + 2x - 10, remainder = 0, k = -2. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 4 Using Synthetic Division to Divide a Third-Degree Polynomial: 4x^2 + 2x - 10.
3. p = -9x^4 + 10x^3 + 7x^2 - 6, d = x - 1 gives quotient = -9x^3 + x^2 + 8x + 8, remainder = 2, row = -9, 1, 8, 8 | 2. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 5 Using Synthetic Division to Divide a Fourth-Degree Polynomial: −9x^3 + x^2 + 8x + 8, remainder 2.
4. p = 6x^3 + 11x^2 - 31x + 15, d = 3x - 2 gives quotient = 2x^2 + 5x - 7, remainder = 1, k = 2/3, row = 6, 15, -21 | 1. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, Example 2 (by long division): 2x^2 + 5x - 7, remainder 1.

## FAQ

### How does synthetic division work?

Write the dividend’s coefficients, highest power first. For a divisor x − k, bring down the first coefficient, multiply it by k, add the product to the next coefficient, and repeat to the end. The last number is the remainder; the others are the quotient’s coefficients, one degree lower.

### What is k for a divisor like x + 2?

k is the number that makes the divisor 0. For x + 2 that is −2, for x − 3 it is 3, and for 3x − 2 it is 2/3.

### What if a power is missing from the dividend?

Write 0 for it. For −9x^4 + 10x^3 + 7x^2 − 6 the coefficients are −9, 10, 7, 0, −6; dividing by x − 1 gives −9x^3 + x^2 + 8x + 8, remainder 2.

### Can I divide by 2x − 1 or 3x − 2?

Yes. Divide by x − k with k = 2/3 for 3x − 2, then divide the quotient (not the remainder) by 3. 6x^3 + 11x^2 − 31x + 15 ÷ (3x − 2) gives 2x^2 + 5x − 7, remainder 1.

### What does the remainder tell me?

By the remainder theorem it is the dividend’s value at k: p(k). A remainder of 0 means x − k is a factor and k is a zero of the polynomial, as with 5x^2 − 3x − 36 and x − 3.

### When can I not use synthetic division?

When the divisor has degree 2 or more, such as x^2 + 1. Use polynomial long division for those; the polynomial calculator does it.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 5.4 Dividing Polynomials: synthetic division, Examples 3, 4 and 5 (5x² − 3x − 36 ÷ (x − 3) = 5x + 12; 4x³ + 10x² − 6x − 20 ÷ (x + 2) = 4x² + 2x − 10; −9x⁴ + 10x³ + 7x² − 6 ÷ (x − 1) = −9x³ + x² + 8x + 8, remainder 2) and Example 2. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials