Polynomial Long Division Calculator

Divide polynomials and see the quotient, remainder, and complete step-by-step long division process.

Enter Polynomial Coefficients

For x³ - 3x + 2, enter: 1, 0, -3, 2 (include 0 for missing terms)
For x - 1, enter: 1, -1

Result

Quotient
x² + x - 2
Remainder: 0
Dividend x³ - 3x + 2
Divisor x - 1
Quotient x² + x - 2
Remainder 0

Step-by-Step Long Division

Dividend = Divisor x Quotient + Remainder

Understanding Polynomial Long Division

Polynomial long division is an algorithm for dividing a polynomial by another polynomial of equal or lower degree. It works similarly to long division with numbers: divide the leading term, multiply, subtract, bring down, and repeat until the remainder has a lower degree than the divisor.

The Division Algorithm

Division Theorem

For polynomials f(x) and g(x) with g(x) not zero:

f(x) = g(x) * q(x) + r(x)

Degree Constraint

The remainder always has lower degree than the divisor.

deg(r) < deg(g) or r = 0

Synthetic Division

A shortcut when dividing by (x - c), using only coefficients.

Faster for linear divisors

How to Perform Polynomial Long Division

  1. Divide: Divide the leading term of the dividend by the leading term of the divisor.
  2. Multiply: Multiply the entire divisor by the result from step 1.
  3. Subtract: Subtract the product from the current dividend.
  4. Bring down: Bring down the next term from the original dividend.
  5. Repeat: Continue until the degree of the remainder is less than the degree of the divisor.

Applications of Polynomial Division

  • Finding roots: If the remainder is zero, the divisor is a factor of the dividend.
  • Partial fractions: Decomposing rational expressions for integration in calculus.
  • Simplification: Reducing complex rational expressions.
  • Factor theorem: Testing whether (x - c) is a factor by checking if f(c) = 0.
  • Remainder theorem: The remainder when dividing f(x) by (x - c) equals f(c).

Tips for Success

  • Always write polynomials in descending order of degree.
  • Include zero coefficients for any missing powers of x.
  • Double-check by multiplying the quotient by the divisor and adding the remainder.
  • Use synthetic division as a shortcut when the divisor is linear (x - c).