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Dividing polynomials worksheets

Polynomial long division is the same three moves as the numeric kind: divide the leading term, multiply the whole divisor by it, subtract. These free printable worksheets ask for the quotient and the remainder, and the answer key writes out every round.

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Dividing polynomials
Dividing polynomials worksheet — Cubic ÷ linear, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The move students drop is the subtraction. They divide the leading term, write it down, and go to the next term of the dividend without taking the product away — which is invisible until the last line comes out wrong. Every round on these keys ends on the polynomial that is actually left, so the place it went wrong is the place it stops matching.

Why this sheet works

Every question is a cubic over a linear factor and the remainder gets a box of its own, which may hold 0. That is how a book asks it — it is the Remainder Theorem’s own phrasing — and it means a sheet can mix the exact divisions with the ones that leave something over without the shape of the answer giving away which is which.

How this one works

One question from this sheet, worked through a step at a time.

(x³ − 8x² + 11x + 20) ÷ (x − 4)

x²− 4x− 5 x − 4x³− 8x²+ 11x+ 20 −x³+ 4x²−4x²+ 11x+ 204x²− 16x−5x+ 205x− 200
  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 − 4) is x³ − 4x² — and take that away, leaving −4x² + 11x + 20. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.
  3. −4x² ÷ x is −4x, so −4x goes in the x place. Multiply the WHOLE divisor by it — −4x(x − 4) is −4x² + 16x — and take that away, leaving −5x + 20. That subtraction is the step that gets skipped, and skipping it is invisible until the last line comes out wrong.
  4. −5x ÷ x is −5, so −5 goes in the constant place. Multiply the WHOLE divisor by it — −5(x − 4) is −5x + 20 — 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 − 4 divides it exactly: (x³ − 8x² + 11x + 20) ÷ (x − 4) = x² − 4x − 5.

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