Polynomial Long Division
Divide one polynomial by another and get the quotient and remainder.
Because the divisor has degree 1, the remainder’s degree is always less than 1.
Polynomial long division splits a dividend into divisor × quotient + remainder. Dividing x³ − 2x² + 4 by x − 1 gives a quotient of x² − x − 1 and a remainder of 3, so x³ − 2x² + 4 = (x − 1)(x² − x − 1) + 3. The remainder’s degree is always less than the divisor’s.
What polynomial long division does
It mirrors the long division you use for whole numbers, but on polynomials. You repeatedly divide the leading term of what is left by the leading term of the divisor, multiply the whole divisor by that piece of the quotient, and subtract. Each subtraction cancels the current leading term and lowers the degree, so the process ends once the leftover — the remainder — has a smaller degree than the divisor.
Enter each polynomial as its coefficients from the highest power down, filling in zeros for any missing term. For x³ − 2x² + 4 the x term is missing, so its coefficients are 1, −2, 0, 4.
Worked example
Divide x³ − 2x² + 4 by x − 1:
- 1 Divide the leading terms. x³ ÷ x = x², the first quotient term.
- 2 Multiply and subtract. x²(x − 1) = x³ − x². Subtract: (x³ − 2x² + 4) − (x³ − x²) = −x² + 4.
- 3 Repeat on the new leading term. −x² ÷ x = −x. −x(x − 1) = −x² + x. Subtract: (−x² + 4) − (−x² + x) = −x + 4.
- 4 Repeat again. −x ÷ x = −1. −1(x − 1) = −x + 1. Subtract: (−x + 4) − (−x + 1) = 3.
- 5 Stop and read off the answer. Degree 0 (the constant 3) is below the divisor’s degree 1, so stop. Quotient x² − x − 1, remainder 3.
The repeating step, each round
Repeat these until the remainder’s degree is less than the divisor’s degree.
| Step | Action | Purpose |
|---|---|---|
| 1. Divide | Leading term of the remainder ÷ leading term of the divisor | Gives the next quotient term |
| 2. Multiply | Multiply the whole divisor by that quotient term | Builds the amount to remove |
| 3. Subtract | Subtract that product from the current remainder | Cancels the leading term, lowers the degree |
| 4. Repeat | Go back to step 1 with the new, smaller remainder | Continues until degree(remainder) < degree(divisor) |
Reading and checking the result
The division identity is your check. Any valid division satisfies dividend = divisor × quotient + remainder. For the worked example, (x − 1)(x² − x − 1) + 3 = (x³ − 2x² + 0x + 1) + 3 = x³ − 2x² + 4, which matches the dividend exactly.
The remainder’s degree is always smaller than the divisor’s. Dividing by a degree-1 divisor leaves a constant remainder; dividing by a degree-2 divisor leaves a remainder of degree 1 or 0. A remainder of 0 means the divisor is a factor.
Synthetic division is the shortcut for a linear divisor. When the divisor is x − c, synthetic division reaches the same quotient and remainder with just the coefficients and c. Long division is the general method — it also handles quadratic and higher-degree divisors.