erweitern (x+y+z)^4

Lösung

erweitern

(x + y + z)^{4}

x^{4} + 4 x^{3} y + 4 x^{3} z + 4 x z^{3} + 4 y z^{3} + 4 x y^{3} + 4 y^{3} z + 6 x^{2} y^{2} + 6 x^{2} z^{2} + 6 y^{2} z^{2} + z^{4} + 12 x^{2} yz + 12 x y z^{2} + 12 x y^{2} z + y^{4}

Schritte zur Lösung

(x + y + z)^{4}

Vereinfache

(x + y + z)^{4} : (x + y + z) (x + y + z) (x + y + z) (x + y + z)

= (x + y + z) (x + y + z) (x + y + z) (x + y + z)

Schreibe

(x + y + z) (x + y + z)

um:

x^{2} + 2 x y + 2 x z + y^{2} + 2 yz + z^{2}

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

Schreibe

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

um:

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

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

Setze Klammern

= x^{3} x + x^{3} y + x^{3} z + 3 x^{2} y x + 3 x^{2} yy + 3 x^{2} yz + 3 x^{2} z x + 3 x^{2} zy + 3 x^{2} zz + 3 x z^{2} x + 3 x z^{2} y + 3 x z^{2} z + 3 y z^{2} x + 3 y z^{2} y + 3 y z^{2} z + 3 x y^{2} x + 3 x y^{2} y + 3 x y^{2} z + z^{3} x + z^{3} y + z^{3} z + 3 y^{2} z x + 3 y^{2} zy + 3 y^{2} zz + 6 x yz x + 6 x yzy + 6 x yzz + y^{3} x + y^{3} y + y^{3} z

Vereinfache

x^{3} x + x^{3} y + x^{3} z + 3 x^{2} y x + 3 x^{2} yy + 3 x^{2} yz + 3 x^{2} z x + 3 x^{2} zy + 3 x^{2} zz + 3 x z^{2} x + 3 x z^{2} y + 3 x z^{2} z + 3 y z^{2} x + 3 y z^{2} y + 3 y z^{2} z + 3 x y^{2} x + 3 x y^{2} y + 3 x y^{2} z + z^{3} x + z^{3} y + z^{3} z + 3 y^{2} z x + 3 y^{2} zy + 3 y^{2} zz + 6 x yz x + 6 x yzy + 6 x yzz + y^{3} x + y^{3} y + y^{3} z : x^{4} + 4 x^{3} y + 4 x^{3} z + 4 x z^{3} + 4 y z^{3} + 4 x y^{3} + 4 y^{3} z + 6 x^{2} y^{2} + 6 x^{2} z^{2} + 6 y^{2} z^{2} + z^{4} + 12 x^{2} yz + 12 x y z^{2} + 12 x y^{2} z + y^{4}

= x^{4} + 4 x^{3} y + 4 x^{3} z + 4 x z^{3} + 4 y z^{3} + 4 x y^{3} + 4 y^{3} z + 6 x^{2} y^{2} + 6 x^{2} z^{2} + 6 y^{2} z^{2} + z^{4} + 12 x^{2} yz + 12 x y z^{2} + 12 x y^{2} z + y^{4}

e x p an d (s + 2) (s + 1)

s im pl i f y (x^{2} + 6)^{2} + 6

e x p an d (x^{2} + x - 2) (3 x + 7)

e x p an d (a^{2} - 2 ab + b^{2}) (4 a^{2} - b^{2})

f a c t or x^{3} + 3 x^{2} - x