How it works
Polynomial long division works like ordinary long division: divide the leading terms to get the next quotient term, multiply that term back through the divisor, subtract, and repeat with what's left until the remaining degree drops below the divisor's.
dividend = (divisor × quotient) + remainder
Worked example
(x² − 3x + 2) ÷ (x − 1).
- Divide leading terms: x² ÷ x = x. Multiply back: x × (x − 1) = x² − x. Subtract: (x² − 3x) − (x² − x) = −2x, bring down +2 → −2x + 2.
- Divide leading terms: −2x ÷ x = −2. Multiply back: −2 × (x − 1) = −2x + 2. Subtract: 0.
Quotient x − 2, remainder 0 — it divides exactly.
Common questions
Why does a remainder of 0 matter?
It means the divisor is a factor of the dividend — the division is exact, with no leftover term. A non-zero remainder means it doesn't divide evenly.
What if the dividend's degree is lower than the divisor's?
The quotient is 0 and the whole dividend is the remainder — there's nothing to divide out, since a smaller-degree polynomial can't contain a factor of higher degree.
Why must I include zero coefficients for missing terms?
Each position in the list corresponds to one specific power of x, in order — skipping a zero would shift every term after it to the wrong power. Writing x² − 4 as [1, 0, -4] keeps the x¹ position explicit even though its coefficient is zero.