Polynomial Division Calculator with Steps
Divide one polynomial by another. Enter the dividend and divisor as text
(e.g. x^3+2x^2-5x+1) or as coefficients, and get the
quotient, the remainder, every long-division step, the synthetic-division
table when the divisor is x − a, and an automatic check that
divisor × quotient + remainder = dividend.
Worked example
(x^3 + 2x^2 − 5x + 1) ÷ (x − 2) — long division:
- x^3 ÷ x = x^2; x^2 × (x − 2) = x^3 − 2x^2; subtract → 4x^2 − 5x.
- 4x^2 ÷ x = 4x; 4x × (x − 2) = 4x^2 − 8x; subtract → 3x + 1.
- 3x ÷ x = 3; 3 × (x − 2) = 3x − 6; subtract → 7.
Quotient: x^2 + 4x + 3 Remainder: 7
✓ Check: (x − 2)(x^2 + 4x + 3) = x^3 + 2x^2 − 5x − 6; plus 7 gives back x^3 + 2x^2 − 5x + 1.
Frequently asked questions
What is polynomial division?
Polynomial division splits one polynomial (the dividend) by another (the divisor) into a quotient plus a remainder, so that divisor × quotient + remainder = dividend, where the remainder's degree is lower than the divisor's. This calculator shows every step of the long-division process and verifies the result.
How do you divide (x^3 + 2x^2 − 5x + 1) by (x − 2)?
Long division gives quotient x^2 + 4x + 3 with remainder 7. Check: (x − 2)(x^2 + 4x + 3) = x^3 + 2x^2 − 5x − 6, and adding the remainder 7 gives back x^3 + 2x^2 − 5x + 1. Because the divisor is monic and linear, synthetic division with a = 2 gives the same answer faster.
When can I use synthetic division instead of long division?
Synthetic division only works when the divisor is a monic linear factor, x − a (for example x − 3 uses a = 3). This tool detects that automatically: it shows the synthetic table for (2x^3 − 6x^2 + 2x − 1) ÷ (x − 3), giving quotient 2x^2 + 2 and remainder 5, and falls back to full long division for anything else, such as division by x^2 + 1.
What happens when the remainder is not zero?
A non-zero remainder is normal: it just means the division is not exact. For (x^4 − 3x^3 + 5x − 2) ÷ (x^2 + 1) the quotient is x^2 − 3x − 1 with remainder 8x − 1, and the tool always verifies that (x^2 + 1)(x^2 − 3x − 1) + (8x − 1) equals the original dividend.
How is this different from a characteristic polynomial calculator?
They share the word 'polynomial' but solve different problems. Polynomial division divides one polynomial by another. A characteristic polynomial calculator computes det(A − λI) of a matrix to find eigenvalues — for that, use our characteristic-polynomial tool instead.
Free tool by NRTechStudio