Polynomial Division Calculator
Divide any polynomial by another polynomial using long division with step-by-step solutions
Dividend ÷ Divisor = Quotient + Remainder/Divisor
Include 0 for missing terms (e.g., 1, 0, -4 for x² - 4)
For (x + 3), enter: 1, 3
Please enter at least 1 coefficient for the dividend.
Please enter at least 1 coefficient for the divisor.
() ÷ ()
Quotient Q(x)
Remainder R(x)
✓ Exact division (no remainder)!
Complete Division Result
= () × () +
Step-by-Step Solution
Step
Multiply:
Final Result: Quotient = , Remainder =
Factor Found!
Since the remainder is 0:
- • () is a factor of the polynomial
- • The polynomial can be factored as: ()()
The Polynomial Long Division Method
Algorithm Steps
- 1. Divide the leading term of dividend by leading term of divisor
- 2. Multiply the entire divisor by this quotient term
- 3. Subtract the result from the current dividend
- 4. Repeat with the new polynomial until remainder degree < divisor degree
Key Concepts
Euclidean Division
A = B × Q + R, where degree(R) < degree(B)
Factorization
If R = 0, then B is a factor of A
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About Polynomial Division Calculator
What is Polynomial Long Division?
Polynomial long division is an algorithm for dividing any polynomial by another polynomial. Unlike synthetic division which only works with linear divisors (x - c), polynomial long division can handle divisors of any degree. The process mirrors traditional long division with numbers.
The Formula Behind the Calculation
Polynomial division follows the Euclidean division formula:
A = B × Q + R
Where:
- A is the dividend (polynomial being divided)
- B is the divisor (polynomial dividing A)
- Q is the quotient (result of division)
- R is the remainder (must have degree less than B, or be 0)
How to Use This Calculator
- Enter Dividend Coefficients: Enter the coefficients of the polynomial you want to divide, in descending order of powers (e.g., "1, -5, 6, 0" for x³ - 5x² + 6x)
- Enter Divisor Coefficients: Enter the coefficients of the divisor polynomial in the same format (e.g., "1, -2" for x - 2)
- Calculate: View the complete step-by-step solution, quotient, and remainder
Understanding Your Results
Quotient Polynomial
The resulting polynomial from division. Its degree equals the dividend degree minus the divisor degree.
Remainder
What's left after division. If the remainder is 0, the divisor is a factor of the dividend.
Complete Division Statement
Shows the relationship: Dividend = Divisor × Quotient + Remainder
FAQ (Frequently Asked Questions)
What if I have missing terms in my polynomial?
Use 0 as a placeholder. For x³ + 4x - 1 (missing x² term), enter: 1, 0, 4, -1
When should I use polynomial vs synthetic division?
Use synthetic division for linear divisors (x - c). Use polynomial long division for quadratic or higher-degree divisors.
How do I verify my answer?
Multiply the quotient by the divisor and add the remainder. You should get the original dividend polynomial.
What if the divisor degree is higher than the dividend?
The quotient will be 0 and the remainder will be the original dividend.
Important Limitations
- Both polynomials must have real coefficients
- The divisor cannot be zero (the zero polynomial)
- Very high degree polynomials may have rounding issues
Disclaimer: This calculator is for educational purposes. Always verify results manually for critical academic work.