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The factor theorem worksheets Grades 10–11

Show x − c is a factor, divide it out, and factor what is left.

The front of the free the factor theorem worksheet
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Free · Worked example · 14 problems · Answer key · US Letter

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How to do it

If P(c) = 0, then x − c is a factor. Divide it out, then factor the quadratic that is left.

x3+x2−10x+8;x+4 is a factor→(x+4)(x−2)(x−1)x^{3} + x^{2} - 10x + 8;\quad x + 4\ \text{is a factor}\quad \to \quad (x + 4)(x - 2)(x - 1)
x3+x2−10x+8=(x+4)(x2−3x+2)x^{3} + x^{2} - 10x + 8 = (x + 4)(x^{2} - 3x + 2)the bottom row is the other factor
(x+4)(x2−3x+2)=(x+4)(x−2)(x−1)(x + 4)(x^{2} - 3x + 2) = (x + 4)(x - 2)(x - 1)factor the quadratic

Example problems

  1. 2x3−8x2−34x+120;x−5 is a factor→2x^{3} - 8x^{2} - 34x + 120;\quad x - 5\ \text{is a factor}\quad \to \quad
  2. x3−21x−20;x−5 is a factor→x^{3} - 21x - 20;\quad x - 5\ \text{is a factor}\quad \to \quad
  3. 2x3−4x2−16x;x+2 is a factor→2x^{3} - 4x^{2} - 16x;\quad x + 2\ \text{is a factor}\quad \to \quad
  4. x3+x2−25x−25;x+1 is a factor→x^{3} + x^{2} - 25x - 25;\quad x + 1\ \text{is a factor}\quad \to \quad

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