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Polynomial division with remainders worksheets

A polynomial division that does not come out exactly is not a mistake, and that is what these sheets are for. These free printable worksheets divide a cubic by a linear factor, and every one of them leaves a remainder.

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Polynomial division with remainders
Polynomial division with remainders worksheet — Cubic ÷ linear, always a remainder, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Students who have only seen exact divisions tend to go back and look for an error when something is left over. A whole page where something is always left over settles that. The answer key writes out each round, so a student can see the last subtraction land on a plain number and stop there.

Why this sheet works

The remainder is also a check. Multiplying the quotient by the divisor and adding the remainder has to give the cubic back, and that is a stronger test than an exact division offers. It is also the first look at the Remainder Theorem: the number left over is the value of the polynomial at the divisor’s root.

How this one works

(x³ − 8x² + 11x + 21) ÷ (x − 4)

x²− 4x− 5 x − 4x³− 8x²+ 11x+ 21 −x³+ 4x²−4x²+ 11x+ 214x²− 16x−5x+ 215x− 201
  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 + 21.
  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 + 21.
  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 1.
  5. 1 is left and there is no x in it to divide, so that is the remainder. (x³ − 8x² + 11x + 21) ÷ (x − 4) = x² − 4x − 5 remainder 1.

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