Exercise

Wir betrachten die Polynome

$f (x) = x^{4} + 2 x^{3} + 6 x^{2} + 2 x + 5$ $g (x) = x^{3} + 5 x^{2} + 11 x + 15$

aus $ℝ [x]$ .

Finde den $\gcd (f (x), g (x))$ und gib ihn in der Form $s (x) f (x) + t (x) g (x)$ mit $s (x), t (x) \in ℝ [x]$ 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: $x^{2} + 2 x + 5$ Linearkombination: $f (x) = (x - 3) g (x) + 10 (x^{2} + 2 x + 5)$ $x^{2} + 2 x + 5 = \frac{1}{10} (f (x) - (x - 3) g (x))$

 (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

$\gcd (f (x), g (x)) = x^{2} + 2 x + 5 = \frac{1}{10} f (x) - (\frac{x}{10} - \frac{3}{10}) g (x)$

Solution

$\gcd (f (x), g (x)) = x^{2} + 2 x + 5 = \frac{1}{10} f (x) - (\frac{x}{10} - \frac{3}{10}) g (x)$

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