Wir betrachten die Polynome

´f(x) = x^4 + 2x^3 + 6x^2 + 2x + 5´ ´g(x) = x^3 + 5x^2 + 11x + 15´

aus ´RR[x]´.

Finde den ´gcd(f(x), g(x))´ und gib ihn in der Form ´s(x)f(x) + t(x)g(x)´ mit ´s(x), t(x) in RR[x]´ an.

Approach
 (x^4 + 2x^3 + 6x^2  +  2x +  5)/(x^3 + 5x^2 + 11x + 15) = x - 3
-(x^4 + 5x^3 + 11x^2 + 15x)
       -3x^3 -  5x^2 - 13x +  5
     -(-3x^3 - 15x^2 - 33x - 45)
               10x^2 + 20x + 50

Normiert: ´x^2 + 2x + 5´ Linearkombination: ´f(x) = (x-3)g(x) + 10(x^2 + 2x + 5)´ ´x^2 + 2x + 5 = 1/10(f(x) - (x-3)g(x))´

 (x^3 + 5x^2 + 11x + 15)/(x^2 + 2x + 5) = x + 3
-(x^3 + 2x^2 +  5x)
        3x^2 + 6x + 15
      -(3x^2 + 6x + 15)
                     0

´gcd(f(x), g(x)) = x^2 + 2x + 5 = 1/10 f(x) - (x/10 - 3/10)g(x)´


Solution

  • ´gcd(f(x), g(x)) = x^2 + 2x + 5 = 1/10 f(x) - (x/10 - 3/10)g(x)´

  • URL:
  • Language:
  • Subjects: math
  • Type: Calculate
  • Duration: 25min
  • Credits: 3
  • Difficulty: 0.6
  • Tags: hpi polynomial gcd
  • Note:
    HPI, 2014-06-30, Mathe 2, Aufgabe 49
  • Created By: adius
  • Created At:
    2014-07-26 19:45:28 UTC
  • Last Modified:
    2014-07-26 19:45:28 UTC