Polynomial Division with Remainder

Calculator for dividing two polynomials producing quotient and remainder

Polynomial Division Calculator

What is calculated?

This function performs the algebraic division of two polynomials and returns the quotient and remainder according to the division principle.

Enter polynomials
Input format

Coefficients can be entered as a sequence of numbers separated by semicolons or spaces.

Example: For \(3x^2+4x+5\) enter "3 4 5" or "3; 4; 5"

Polynomial long division

As with numeric division: \(P(x) = Q(x) \cdot S(x) + R(x)\) with quotient S(x) and remainder R(x)


÷
Result
P(x) =
Q(x) =
Quotient S(x) =
Remainder R(x) =
Verification: P(x) = Q(x) · S(x) + R(x)

Polynomial Division Info

Division principle

Like numeric division:

  • Dividend ÷ Divisor = Quotient + Remainder
  • P(x) = Q(x) · S(x) + R(x)
  • deg(R) < deg(Q)
  • Apply long division

Tip: The degree of the remainder is always smaller than the degree of the divisor.

Division algorithm
1. Divide the highest power terms
2. Multiply by the divisor
3. Subtract from the dividend
4. Bring down the next term
5. Repeat until finished
Important

The divisor must not be the zero polynomial. The degree of the divisor should be ≤ degree of the dividend.

Formulas for polynomial division

Division principle
\[P(x) = Q(x) \cdot S(x) + R(x)\] Dividend = Divisor × Quotient + Remainder
Degree condition
\[\deg(R) < \deg(Q)\] Degree of remainder < degree of divisor
Uniqueness
\[S(x) \text{ and } R(x) \text{ are unique}\] Quotient and remainder are unique
Verification
\[Q(x) \cdot S(x) + R(x) = P(x)\] Check by recomputing

Detailed example

Example: (3x² + 4x + 5) ÷ (2x + 3)
    1.5x + 0.25
   _______________
2x+3 | 3x² + 4x + 5
       3x² + 4.5x      ← 1.5x × (2x + 3)
       ___________
            -0.5x + 5
            -0.5x - 0.75  ← 0.25 × (2x + 3)
            ____________
                   5.75   ← Remainder

1First division

\[\frac{3x^2}{2x} = 1.5x\]

Divide highest terms

2Multiplication

1.5x × (2x + 3) = 3x² + 4.5x

Multiply quotient term by divisor

3Subtraction

\[(3x^2 + 4x) - (3x^2 + 4.5x) = -0.5x\]

Subtract from dividend

4Bring down next term

\[-0.5x + 5\]

Bring down constant term

5Second division

\[\frac{-0.5x}{2x} = -0.25\]

Next quotient term

6Final result

Quotient: \(1.5x - 0.25\)
Remainder: \(5.75\)
Verification
\[(2x + 3) \cdot (1.5x - 0.25) + 5.75 = 3x^2 + 4x + 5\]

Check confirms the result ✓

Division algorithm

Step-by-step instructions
  1. Arrange: Sort polynomials by descending powers
  2. Divide: Divide the highest term of the dividend by the highest term of the divisor
  3. Multiply: Multiply the result by the entire divisor
  4. Subtract: Subtract the product from the current dividend
  5. Repeat: Continue with the new dividend
  6. Stop: When degree of remainder < degree of divisor
Practical tips
  • Fill gaps: Insert zero coefficients for missing powers
  • Be careful: Pay attention to signs when subtracting
  • Check: Verify result by recomputing
  • Remainder check: Degree of remainder must be smaller
Common mistakes
  • Forgetting signs when subtracting
  • Not accounting for missing terms
  • Stopping the division too early

Polynomial Functions

Add  •  Scalar Divide  •  Divide Pointwise  •  Divide with Remainder  •  Multiply  •  Pointwise Multiply  •  Scalar Multiply  •  Subtract  •