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Free polynomial long division calculator & polynomial division calculator. Divide polynomials of any degree with step-by-step solutions, complete work shown, and remainder analysis. Perfect for algebra students learning polynomial division.
Last updated: February 2, 2026
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Enter dividend polynomial like x^3 + 2x^2 - 5x + 3
Enter divisor polynomial like x^2 - 1 or x + 2
Quotient:
x + 4
Remainder:
7
Step-by-Step Solution:
Dividing: (x^3 + 2x^2 - 5x + 3) ÷ (x^2 - 1)
Divisor leading term: 1x^2
Step 1: Divide x^3 by x^2 = x
Subtract: 2x^2 - 4x + 3
Step 2: Divide 2x^2 by x^2 = 2
Subtract: -4x + 1
Complete Result:
Dividend ÷ Divisor = x + 4 + (7)/(divisor)
Long Division Tips:
Educational feature
Detailed Work Shown
See every division, multiplication, and subtraction step explained
Versatility
Any Degree Polynomials
Works with quadratic, cubic, quartic, and higher degree divisors
Results format
Q(x) + R(x)/D(x)
Express results as quotient plus remainder over divisor
Classic method
Standard Algorithm
Uses the traditional long division format familiar from arithmetic
Learning aid
All Steps Visible
Perfect for checking homework and understanding the process
Division Algorithm
f(x) = d(x)·q(x) + r(x)
Complete division following the polynomial Division Algorithm theorem
Divide (x³ + 2x² - 5x + 3) by (x² - 1) using polynomial long division:
Quotient
x + 2
Remainder
-4x + 5
Result: (x³ + 2x² - 5x + 3) ÷ (x² - 1) = (x + 2) + (-4x + 5)/(x² - 1)
Our polynomial long division calculator implements the Division Algorithm for polynomials, which states that for polynomials f(x) (dividend) and d(x) (divisor with d(x) ≠ 0), there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that f(x) = d(x)·q(x) + r(x), where deg(r) < deg(d).
Step 1: Arrange polynomials in descending order by degree
Step 2: Divide leading term of dividend by leading term of divisor
Step 3: Multiply entire divisor by the result from step 2
Step 4: Subtract this product from the dividend
Step 5: Bring down the next term (if any)
Step 6: Repeat steps 2-5 until deg(remainder) < deg(divisor)
This algorithm mirrors the long division process for numbers but operates on polynomials. Each iteration reduces the degree of the remaining polynomial until we can no longer divide.
Visual representation of the long division layout and process
Polynomial long division is based on the Division Algorithm theorem, a fundamental result in algebra. The theorem guarantees that division of polynomials always produces a unique quotient and remainder with the remainder having a lower degree than the divisor. This property makes polynomial division well-defined and predictable.
Need help with other polynomial operations? Check out our synthetic division calculator and quadratic formula calculator.
Get Custom Calculator for Your PlatformResult:
Quotient: 2x² + x - 3
Remainder: 8x + 1
(2x⁴ - 3x³ + x² + 5x - 2) = (x² - 2x + 1)(2x² + x - 3) + (8x + 1)
Divide x³ - 8 by x - 2
Result: x² + 2x + 4, Remainder: 0
f(x) = (x³ + 1)/(x² - 1)
Quotient x is the oblique asymptote
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