Synthetic Division Calculator with Steps and Graph

Calculator Form

Polynomial coefficients

Enter highest-degree to constant term. Fractions like 1/2 are allowed.

Divisor x coefficient (a)

Use ax + b. Example: a = 1 for x - 2.

Divisor constant (b)

For x - 2, enter b = -2.

Decimal precision

Graph minimum x

Graph maximum x

Graph points

Reset

Example Data Table

Polynomial	Divisor	Synthetic Constant	Quotient	Remainder
2x^3 - 3x^2 + 0x + 5	x - 2	2	2x^2 + x + 2	9
x^3 - 6x^2 + 11x - 6	x - 1	1	x^2 - 5x + 6	0
3x^4 + 2x^3 - x + 8	2x + 4	-2	1.5x^3 - 2x^2 + 4x - 8.5	42

Formula Used

Synthetic division is a compact version of polynomial long division for linear divisors. Write the divisor as ax + b and convert it to the synthetic constant c = -b / a.

Start with b₀ = a₀ Then bₖ = aₖ + c \cdot bₖ₋₁ for each next coefficient Remainder = last running value Quotient for x - c = previous running values Quotient for ax + b = previous running values divided by a

The remainder theorem also applies: when the divisor is x - c, the remainder equals P(c). For ax + b, the same synthetic constant c = -b / a gives the same remainder value.

How to Use This Calculator

Enter polynomial coefficients from highest power to constant term.
Enter the divisor in the form ax + b using the two divisor fields.
Select the decimal precision you want in the output.
Set the graph range and number of sample points.
Press the calculate button to show the result section above the form.
Review the quotient, remainder, step table, and graph.
Use the export buttons to download the computed steps as CSV or PDF.

Frequently Asked Questions

1. What is synthetic division?

Synthetic division is a faster method for dividing a polynomial by a linear divisor. It replaces most subtraction steps from long division with repeated multiplication and addition.

2. When should I use this method?

Use it when the divisor is linear, such as x - 3 or 2x + 4. It is especially useful for factor testing, root checking, and simplifying repeated polynomial divisions.

3. Why do I enter coefficients instead of the full polynomial?

Coefficient input keeps the calculator simple and precise. It also supports missing powers by letting you enter zero directly, such as 4, 0, -2, 7.

4. Can I divide by 2x + 3?

Yes. Enter a = 2 and b = 3. The calculator converts the divisor to its synthetic constant, computes the row, and rescales the quotient correctly.

5. What does a remainder of zero mean?

A zero remainder means the divisor is an exact factor of the polynomial. In that case, the corresponding root satisfies the polynomial exactly.

6. Why is the graph helpful?

The plot helps you inspect the original polynomial visually, compare it with the quotient, and see the function value at the synthetic constant used in the division.

7. Can I use decimal or fractional coefficients?

Yes. You can enter decimals like 2.5 or fractions like 3/4. The calculator converts them to numeric values before performing the synthetic division steps.

8. How is this different from long division?

Long division works for broader divisor types, while synthetic division is streamlined for linear divisors. It is shorter, faster, and easier to check by hand.