faktorisieren (x+y+z)^3-x^3-y^3-z^3

Lösung

faktorisieren

(x + y + z)^{3} - x^{3} - y^{3} - z^{3}

3 (x^{2} y + x^{2} z + x y^{2} + x z^{2} + 2 x yz + y z^{2} + y^{2} z)

Schritte zur Lösung

(x + y + z)^{3} - x^{3} - y^{3} - z^{3}

(x + y + z)^{3} = (x + y + z) (x + y + z) (x + y + z)

= (x + y + z) (x + y + z) (x + y + z) - x^{3} - y^{3} - z^{3}

Multipliziere aus

(x + y + z) (x + y + z) (x + y + z) : x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z

= x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z - x^{3} - y^{3} - z^{3}

Vereinfache

x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z - x^{3} - y^{3} - z^{3} : 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + 3 y z^{2} + 3 y^{2} z

= 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + 3 y z^{2} + 3 y^{2} z

Schreibe um

= 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 3 \cdot 2 x yz + 3 y z^{2} + 3 y^{2} z

Klammere gleiche Terme aus

3

= 3 (x^{2} y + x^{2} z + x y^{2} + x z^{2} + 2 x yz + y z^{2} + y^{2} z)

9 x^{2} - 42 x + 49 = 0

s im pl i f y (\frac{6}{7})^{0}

x + \frac{x}{3} \leq 20

4 x^{2} = 13 x + 35

x + 5 = - 3 x + 25