Compute in a polynomial ring

Let p(x) = 2x^3 - 3x^2 + 4x - 5 and let q(x) = 7x^3 + 33x - 4. In each of parts (a), (b), and (c), compute p+q and pq under the assumption that the coefficients of the two given polynomials are taken from the specified ring (where the integer coefficients are taken mod n in parts (b) and (c)): (a) R = \mathbb{Z}, (b) R = \mathbb{Z}/(2), R = \mathbb{Z}/(3).


  1. Evidently, p(x)+q(x) = 9x^3 - 3x^2 + 37x - 9 and p(x)q(x) = 14x^6 - 21x^5 + 94x^4 - 142 x^3 + 144 x^2 - 181 x + 20.
  2. In \mathbb{Z}/(2), p(x) = x^2 + 1 and q(x) = x^3 + x. Evidently, P(x) + q(x) = x^3 + x^2 + x + 1 while p(x)q(x) = x^5 + x.
  3. In \mathbb{Z}/(3), p(x) = 2x^3 + x + 1 and q(x) = x^3 + 2. Evidently, p(x) + q(x) = x while p(x)q(x) = 2x^6 + x^4 + + 2x^3 2x + 2.
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Comments

  • Danny  On November 26, 2011 at 12:37 am

    can you please tell me why in part 3, the ciefficient of x^3 in q is 2? I thought it`s 1.

    • nbloomf  On November 28, 2011 at 10:34 am

      That was a mistake on my part. Thanks!

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