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Dividing polynomials, step by step

Polynomial long division is the same three moves as the numeric kind.

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The question

Divide: (x³ + 8x² + 10x − 12) ÷ (x + 6)

(x³ + 8x² + 10x − 12) ÷ (x + 6)

x²+ 2x− 2 x + 6x³+ 8x²+ 10x− 12 −x³− 6x²2x²+ 10x− 12−2x²− 12x−2x− 122x+ 120
  1. A polynomial goes into the house the same way a number does, and long division starts at the LEFT. Every term sits in its own column, and that is what makes the subtractions line up.
  2. x³ ÷ x is x², so x² goes in the x² place. Multiply the WHOLE divisor by it — x²(x + 6) is x³ + 6x² — and take that away, leaving 2x² + 10x − 12. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.
  3. 2x² ÷ x is 2x, so 2x goes in the x place. Multiply the WHOLE divisor by it — 2x(x + 6) is 2x² + 12x — and take that away, leaving −2x − 12. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.
  4. −2x ÷ x is −2, so −2 goes in the constant place. Multiply the WHOLE divisor by it — −2(x + 6) is −2x − 12 — and take that away, leaving 0. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.
  5. Nothing is left, so x + 6 divides it exactly: (x³ + 8x² + 10x − 12) ÷ (x + 6) = x² + 2x − 2.

How it works.

  1. 01

    A polynomial goes into the house the same way a number does, and long division starts at the LEFT. Every term sits in its own column, and that is what makes the subtractions line up.

  2. 02

    x³ ÷ x is x², so x² goes in the x² place. Multiply the WHOLE divisor by it — x²(x + 6) is x³ + 6x² — and take that away, leaving 2x² + 10x − 12. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.

  3. 03

    2x² ÷ x is 2x, so 2x goes in the x place. Multiply the WHOLE divisor by it — 2x(x + 6) is 2x² + 12x — and take that away, leaving −2x − 12. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.

  4. 04

    −2x ÷ x is −2, so −2 goes in the constant place. Multiply the WHOLE divisor by it — −2(x + 6) is −2x − 12 — and take that away, leaving 0. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.

  5. 05

    Nothing is left, so x + 6 divides it exactly: (x³ + 8x² + 10x − 12) ÷ (x + 6) = x² + 2x − 2.

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