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Polynomial long division worksheets Grades 10–11

Divide polynomials with long division and write the quotient and remainder.

The front of the free polynomial long division worksheet
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Free · Worked example · 10 problems · Answer key · US Letter

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How to do it

Divide the leading terms, multiply, subtract, and repeat. Check: divisor × quotient + remainder = dividend.

(3x3−x2+2x−4)÷(x−1)=3x2+2x+4(3x^{3} - x^{2} + 2x - 4) \div (x - 1) = 3x^{2} + 2x + 4
3x3÷x=3x2;subtract 3x2(x−1)=3x3−3x2:2x2+2x−43x^{3} \div x = 3x^{2};\quad \text{subtract }3x^{2}(x - 1) = 3x^{3} - 3x^{2}:\quad 2x^{2} + 2x - 4
2x2÷x=2x;subtract 2x(x−1)=2x2−2x:4x−42x^{2} \div x = 2x;\quad \text{subtract }2x(x - 1) = 2x^{2} - 2x:\quad 4x - 4
4x÷x=4;subtract 4(x−1)=4x−4:04x \div x = 4;\quad \text{subtract }4(x - 1) = 4x - 4:\quad 0
(x−1)(3x2+2x+4)=3x3−x2+2x−4(x - 1)(3x^{2} + 2x + 4) = 3x^{3} - x^{2} + 2x - 4check: divisor × quotient + remainder

Example problems

  1. (x3+4x2+x−14)÷(x+2)=(x^{3} + 4x^{2} + x - 14) \div (x + 2) =
  2. (6x3+3x2−3x+11)÷(2x+3)=(6x^{3} + 3x^{2} - 3x + 11) \div (2x + 3) =
  3. (x3−x2−10x−8)÷(x+2)=(x^{3} - x^{2} - 10x - 8) \div (x + 2) =
  4. (3x3−5x2+9x−10)÷(x2+3)=(3x^{3} - 5x^{2} + 9x - 10) \div (x^{2} + 3) =

Quadratic and polynomial functions

  1. Vertex form of a quadratic
  2. Finding the vertex from standard form
  3. The discriminant
  4. Quadratic equations with complex roots
  5. Polynomial long division
  6. Synthetic division
  7. The remainder theorem
  8. The factor theorem
  9. The rational root theorem