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Polynomial Division

Polynomial Division Calculator

How to Divide Polynomials

Polynomial division is similar to numerical long division. It involves dividing one polynomial (the dividend) by another (the divisor) to obtain a quotient and possibly a remainder. Here's the step-by-step process:

  1. Arrange Polynomials

    Write both polynomials in standard form (highest degree first). If any degrees are missing, include them with 0 coefficients.

  2. Divide Leading Terms

    Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.

  3. Multiply and Subtract

    Multiply the entire divisor by the new quotient term and subtract this from the dividend.

  4. Bring Down Next Term

    Bring down the next term from the original dividend and repeat the process until the remainder has a lower degree than the divisor.

  5. Write Final Result

    The final answer is the quotient plus any remainder over the divisor.

Example:

Divide (x³ - 12x² - 42) by (x - 3) Step 1: Arrange polynomials Dividend: x³ - 12x² + 0x - 42 Divisor: x - 3 Step 2: Divide leading terms x³ ÷ x = x² (first term of quotient) Step 3: Multiply and subtract x² × (x - 3) = x³ - 3x² Subtract from dividend: (-12x² + 3x²) = -9x² Bring down next term: -9x² + 0x Step 4: Repeat process -9x² ÷ x = -9x (next term of quotient) -9x × (x - 3) = -9x² + 27x Subtract: (0x - 27x) = -27x Bring down next term: -27x - 42 Step 5: Final division -27x ÷ x = -27 (final term of quotient) -27 × (x - 3) = -27x + 81 Subtract: (-42 - 81) = -123 (remainder) Final Result: x² - 9x - 27 with remainder -123 or: x² - 9x - 27 - 123/(x - 3)