Polynomial Division and Factoring: Practice Exercises

Here are various Exercises to accompany the section Polynomial Division and Factoring.  

Polynomial Division

  1. Use polynomial long-division to compute the quotient, q, and remainder, r, after dividing p by d for each of the following pairs of polynomials.  Check your answer by plugging your answers into the equation p = d· q + r and simplifying.
    1. Divide p(x) = x3 - x2 + 3x - 9 by  d(x) = x - 2.
    2. Divide p(x) = 3x4 + 2x2 - 5x + 1 by  d(x) = x2 + 3x - 2.
    3. Divide p(x) = 2x2 + 6x - 9 by  d(x) = 3x - 1.
    4. Pick your own polynomial, p, and a lower degree polynomial, d, and divide d into p to find the quotient, q, and remainder, r.

      Repeat this Exercise as often as necessary until you are confident in your ability to divide polynomials.

    Solution.

Likely Factors of Rational Polynomials and Rational Roots

  1. Use the Rational Root Theorem to list all possible rational roots and corresponding integral, linear factors of the following polynomials.  
    1. x3 - 4x2 + 3x - 15
    2. 4x3 - 3x2 + x - 15

    Solution.

  2. Use our strategy for factoring rational polynomials to factor each of the following polynomials as much  as possible (i.e., find as many rational roots as possible).  To help you narrow your search, some values of each polynomial are already given.
    1. Factor p(x) = 2x3 - 3x2 - 9x + 10.  Hint:
      x p(x)
      -9
      -6
      -3
      0
      3
      6
      9      		 -1610
      -476
      -44
      10
      10
      280
      1144
    2. Factor q(x) = 4x3 + 6x2 + 2x + 3.  Hint:
      x p(x)
      -3
      -1.8
      -0.6
      0.6
      1.8
      3      		 -57
      -4.488
      3.096
      7.224
      49.368
      171
    3. Factor r(x) = 60x4 - 40x3 - 5x2 + 5x.  Hint:
      x p(x)
      -1
      -0.6
      -0.2
      0.2
      0.6
      1      		 90
      11.616
      -0.784
      0.576
      0.336
      20

    Solution.

Factoring Over the Real and Complex Numbers and Prime Polynomials

  1. For each of the following polynomials, factor them as much as possible over each of the following number systems: 
    1. Rational numbers,
    2. Real numbers, and
    3. Complex numbers.
    1. p(x) = -2x3 - 4x2 + 4x
    2. q(x) = -2x3 - 4x2 - 10x
    3. r(x) = x3 - 1
    4. t(x) = x3 - 2

    Solution.

Go to Sketching Rational Functions.

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