Polynomial Long Division Calculator
Long Division · Synthetic Division · Remainder Theorem · Factor Check
Divide polynomial P(x) by divisor D(x). Get quotient Q(x) and remainder R(x) with P(x) = Q(x)·D(x) + R(x). Full step-by-step visual layout.
Quick Examples
P(x) = —
D(x) = —
Remainder Theorem & Factor Check
Long Division — Step-by-Step Layout
Synthetic Division Table
Row 1: original coefficients | Row 2: carry-down products | Row 3: sums (quotient coefficients + remainder)
Step-by-Step Explanation
What Is Polynomial Long Division?
Polynomial long division is the standard algorithm for dividing one polynomial by another, producing a quotient and a remainder. Just as dividing integers gives a result like 17 ÷ 5 = 3 remainder 2 (meaning 17 = 3 × 5 + 2), dividing polynomials gives P(x) = Q(x) · D(x) + R(x), where R(x) has a lower degree than D(x).
This operation is fundamental in algebra and appears in many areas of mathematics — from factoring polynomials and finding roots, to partial fractions in calculus, to error-correcting codes in computer science.
The Long Division Algorithm Step by Step
To divide polynomial P(x) by D(x):
- Divide the leading term of P(x) by the leading term of D(x) to get the first quotient term q₁.
- Multiply q₁ by the entire divisor D(x).
- Subtract this product from P(x) to obtain a new, lower-degree remainder.
- Repeat the process with the new remainder as the new dividend, until the degree of the remainder falls below the degree of D(x).
- The accumulated quotient terms form Q(x); the final remainder is R(x).
Example: Dividing (2x² − x − 6) by (x − 2):
| Step | Operation | Current Remainder |
|---|---|---|
| 1 | 2x² ÷ x = 2x (first quotient term) | 2x² − x − 6 |
| 2 | 2x × (x − 2) = 2x² − 4x (subtract) | 3x − 6 |
| 3 | 3x ÷ x = 3 (second quotient term) | 3x − 6 |
| 4 | 3 × (x − 2) = 3x − 6 (subtract) | 0 |
Result: Q(x) = 2x + 3, R = 0. Since R = 0, (x − 2) is a factor of 2x² − x − 6.
Synthetic Division — The Shortcut for Linear Divisors
Synthetic division is a streamlined technique for dividing any polynomial by a linear divisor of the form (x − k). Instead of writing out polynomial expressions at each step, it works only with the coefficients in a compact three-row table:
- Row 1: Coefficients of the dividend (include zeros for missing degree terms)
- Row 2: Products computed during division (shifted right)
- Row 3: Sums — these are the quotient coefficients (last entry is the remainder)
Synthetic division is faster than long division and is the preferred method when dividing by (x − k). However, it cannot be used for divisors of degree 2 or higher — you must use long division in those cases.
The Remainder Theorem
The Remainder Theorem states: when P(x) is divided by (x − k), the remainder R equals P(k). This is a powerful result because it means you can evaluate the remainder without performing any division at all — just substitute k into P(x). Our calculator automatically states this whenever your divisor is linear.
The Factor Theorem
The Factor Theorem follows directly from the Remainder Theorem: (x − k) is a factor of P(x) if and only if P(k) = 0. In other words, if the division yields remainder 0, then k is a root of the polynomial and (x − k) divides it exactly. This is how polynomial factoring works — you test candidate roots and factor them out one at a time.
Applications
- Factoring polynomials: Use the Rational Root Theorem to identify candidate roots, then synthetic division to confirm them and reduce the degree.
- Finding all roots: Once a linear factor is confirmed, divide it out to obtain a lower-degree quotient, then find its roots recursively.
- Partial fraction decomposition: When the numerator degree ≥ denominator degree, long division must be performed first to extract the polynomial part before applying partial fractions for integration.
- Simplifying rational expressions: Division allows cancellation of common factors in rational functions.
- Error-correcting codes: Reed-Solomon and CRC codes use polynomial division over finite fields.