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العربية

vereinfachen (x+y+z)^3-(x-y-z)^3-6(x+y+z)(x-y-z)(y+z)

Lösung

vereinfachen

(x + y + z)^{3} - (x - y - z)^{3} - 6 (x + y + z) (x - y - z) (y + z)

24 y z^{2} + 8 z^{3} + 24 y^{2} z + 8 y^{3}

Schritte zur Lösung

(x + y + z)^{3} - (x - y - z)^{3} - 6 (x + y + z) (x - y - z) (y + z)

Schreibe

(x + y + z)^{3}

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)^{3} - 6 (x + y + z) (x - y - z) (y + z)

Schreibe

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

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^{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} - 6 (x + y + z) (x - y - z) (y + z)

Schreibe

- 6 (x + y + z) (x - y - z) (y + z)

um:

- 6 x^{2} y + 18 y^{2} z + 18 y z^{2} + 6 y^{3} + 6 z^{3} - 6 x^{2} z

= 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} - 6 x^{2} y + 18 y^{2} z + 18 y z^{2} + 6 y^{3} + 6 z^{3} - 6 x^{2} z

Fasse gleiche Terme zusammen

= x^{3} - x^{3} + 3 x^{2} y + 3 x^{2} y - 6 x^{2} y + 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

x^{3} - x^{3} = 0

= 3 x^{2} y + 3 x^{2} y - 6 x^{2} y + 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 x^{2} y + 3 x^{2} y - 6 x^{2} y = 0

= 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 x^{2} z + 3 x^{2} z - 6 x^{2} z = 0

= 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 x z^{2} - 3 x z^{2} = 0

= 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 y z^{2} + 3 y z^{2} + 18 y z^{2} = 24 y z^{2}

= 24 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 x y^{2} - 3 x y^{2} = 0

= 24 y z^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

z^{3} + 6 z^{3} + z^{3} = 8 z^{3}

= 24 y z^{2} + 8 z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

3 y^{2} z + 3 y^{2} z + 18 y^{2} z = 24 y^{2} z

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

6 x yz - 6 x yz = 0

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + y^{3} + 6 y^{3} + y^{3}

Addiere gleiche Elemente:

y^{3} + 6 y^{3} + y^{3} = 8 y^{3}

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + 8 y^{3}

s im pl i f y (m^{8} n^{4})^{4}

e x p an d (1 - a)^{8}

s im pl i f y (\frac{( a ^{2} )}{b})^{7} \cdot (\frac{( a ^{3} \cdot b ^{2} )}{( c ^{3} )})^{4}

s im pl i f y f (- 1) + 5 f (0) - f (v^{2})

e x p an d (x + 1) \cdot x