How Polynomial Division works
Polynomial long division divides a dividend polynomial A(x) by a divisor polynomial B(x) to produce a quotient Q(x) and a remainder R(x) such that A(x) = B(x) · Q(x) + R(x), with the degree of R(x) strictly less than the degree of B(x). It mirrors ordinary numeric long division, but instead of dividing digits you repeatedly divide, multiply, and subtract leading terms of polynomials.
The long division algorithm, step by step
- Divide the leading term of the current remainder by the leading term of the divisor to get the next quotient term.
- Multiply the entire divisor by that quotient term and subtract the result from the current remainder, canceling the leading term.
- Repeat with the new, lower-degree remainder until its degree is smaller than the divisor's degree — what's left is the final remainder R(x).
Understanding the output
- Quotient and remainder: the quotient Q(x) has degree deg(A) − deg(B); the remainder R(x) always has degree less than deg(B) (or is 0).
- Exact division: if R(x) = 0, the divisor is a factor of the dividend and A(x) = B(x) · Q(x) exactly.
- Checking your answer: multiply B(x) by Q(x) and add R(x) back — the result must equal the original dividend A(x). This catches arithmetic and sign errors.
- Missing powers: if a term is absent (e.g., no x² term), enter 0 as its coefficient so the place values line up correctly.
Synthetic division shortcut
When the divisor is a linear binomial of the form x − c, synthetic division gives the same quotient and remainder as long division but with less writing. This calculator performs the general long-division algorithm, so it also works for quadratic, cubic, or higher-degree divisors where synthetic division does not apply.
Frequently Asked Questions
What does polynomial long division actually compute?
It finds a quotient Q(x) and a remainder R(x) so that Dividend(x) = Divisor(x) × Q(x) + R(x), with the degree of R(x) always less than the degree of the divisor. It works the same way as numeric long division, but you divide, multiply, and subtract leading terms of polynomials instead of digits.
How do I enter a polynomial that is missing a term, like x^3 + 1?
Include a 0 for the missing power. For x^3 + 1 (no x^2 or x term), enter coefficients from the highest degree down as 1, 0, 0, 1, which the calculator reads as 1x^3 + 0x^2 + 0x + 1.
What does it mean if the remainder is 0?
A remainder of 0 means the divisor divides the dividend evenly — the divisor is a factor of the dividend, and Dividend(x) = Divisor(x) × Q(x) exactly with no leftover term.
How is this different from synthetic division?
Synthetic division is a shortcut that only works when the divisor is a linear binomial of the form x − c. This calculator performs full polynomial long division, so the divisor can have any degree.