Polynomial Long Division
Long division divides one polynomial by another and returns a quotient and a remainder, one term at a time, just like long division with numbers.
The method
- Divide the leading term of the dividend by the leading term of the divisor. That is the next term of the quotient.
- Multiply the whole divisor by that term and subtract the result.
- Repeat with what is left until its degree is lower than the divisor's.
Worked example
Divide 2x³ + 3x² − 5x + 6 by x + 2.
- 2x³ ÷ x = 2x². Subtract 2x³ + 4x² to leave −x² − 5x.
- −x² ÷ x = −x. Subtract −x² − 2x to leave −3x + 6.
- −3x ÷ x = −3. Subtract −3x − 6 to leave 12.
The quotient is 2x² − x − 3 with remainder 12. Check: (x + 2)(2x² − x − 3) + 12 = 2x³ + 3x² − 5x − 6 + 12 = 2x³ + 3x² − 5x + 6. ✓
Common mistake
Forgetting to subtract every term of the product, especially signs, or skipping a missing power: write 0x² as a placeholder.
Where it is used
Long division factors polynomials once you know a root, and it exposes the slant asymptote of a rational function. Missing powers need placeholder zeros: dividing x³ − 1 by x − 1, write x³ + 0x² + 0x − 1. The quotient is x² + x + 1 with remainder 0, because (x − 1)(x² + x + 1) = x³ − 1. A zero remainder means the divisor is a factor.