Polynomial Long Division Calculator
Divide polynomials with step-by-step synthetic and long division logic to find the quotient and remainder.
How to Use the Polynomial Long Division Calculator
Polynomial long division is an algebraic algorithm for dividing a polynomial by another polynomial of the same or lower degree. Just as with numerical long division in elementary arithmetic, polynomial division yields a quotient and a remainder, fulfilling the division identity P(x) = D(x)Q(x) + R(x).
Formula and Calculation Methodology
- P(x): Dividend polynomial: a₃x³ + a₂x² + a₁x + a₀
- x – r: Linear divisor factor with root r
- Q(x): Quotient polynomial resulting from division
- R: Remainder constant according to the Polynomial Remainder Theorem
When dividing by a linear factor of the form (x – r), synthetic division provides a streamlined, rapid shortcut by manipulating only the numerical coefficients. By the Polynomial Remainder Theorem, the remainder of dividing P(x) by (x – r) is precisely equal to evaluating the polynomial directly at x = r: if R = 0, then (x – r) is a confirmed algebraic root and factor of P(x).
Polynomial Division & Rational Function Asymptote Applications
| Dividend Degree | Divisor Degree | Quotient Degree | Asymptote Behavior |
|---|---|---|---|
| Degree 3 (Cubic) | Degree 1 (Linear) | Degree 2 (Quadratic) | Parabolic Asymptote |
| Degree 2 (Quadratic) | Degree 1 (Linear) | Degree 1 (Linear) | Slant / Oblique Asymptote |
| Degree n | Degree n | Degree 0 (Constant) | Horizontal Asymptote |
| Degree n | Degree n + 1 | 0 (Proper Fraction) | Horizontal Asymptote at y = 0 |
Frequently Asked Questions
What is the difference between polynomial long division and synthetic division?
Polynomial long division can divide by polynomials of any degree (e.g. dividing by x² + 2x + 5), whereas synthetic division is an optimized shorthand specifically designed for linear divisors of the form (x – r).
How do I find a slant (oblique) asymptote using polynomial division?
When the degree of the numerator polynomial is exactly one greater than the degree of the denominator polynomial, dividing the numerator by the denominator gives a linear quotient (mx + b), which represents the slant asymptote.
What does it mean if the remainder is 0?
If the remainder is zero, the divisor divides the dividend evenly without fractions, proving that the divisor is an exact factor of the polynomial and its root is an x-intercept of the function.