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The rational root theorem worksheets Grades 10–11

List the possible rational roots, find one that works, divide and factor.

The front of the free the rational root theorem worksheet
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Free · Worked example · 8 problems · Answer key · US Letter

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How to do it

Possible rational roots are ± a factor of the constant over a factor of the leading coefficient. Find one that works, divide, and factor.

x3+4x2−9x−36=0x=−4 or x=−3 or x=3x^{3} + 4x^{2} - 9x - 36 = 0\quad x = -4\ \text{or }x = -3\ \text{or }x = 3
try ±1,±2,±3,±4,±6,±9,±12,±18,±36\text{try }\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36factors of the constant over factors of the leading coefficient
(x+4)(x2−9)=0(x + 4)(x^{2} - 9) = 0the bottom row is the other factor
(x+4)(x−3)(x+3)=0(x + 4)(x - 3)(x + 3) = 0factor the quadratic
x=−4 or x=−3 or x=3x = -4\ \text{or }x = -3\ \text{or }x = 3

Example problems

  1. 2x3+13x2+13x−10=02x^{3} + 13x^{2} + 13x - 10 = 0
  2. x3−19x+30=0x^{3} - 19x + 30 = 0
  3. 3x3−x2−12x+4=03x^{3} - x^{2} - 12x + 4 = 0
  4. 3x3−26x2+51x+20=03x^{3} - 26x^{2} + 51x + 20 = 0

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