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Beliebt Algebra >

faktorisieren x^3y^6z^{12}-t^{12}

Lösung

faktorisieren

x^{3} y^{6} z^{12} - t^{12}

Lösung

(x y^{2} z^{4} - t^{4}) (x^{2} y^{4} z^{8} + x y^{2} z^{4} t^{4} + t^{8})

Schritte zur Lösung

x^{3} y^{6} z^{12} - t^{12}

Schreibe

x^{3} y^{6} z^{12}

um:

(x y^{2} z^{4})^{3}

= (x y^{2} z^{4})^{3} - t^{12}

Wende Exponentenregel an:

a^{b c} = (a^{b})^{c}

t^{12} = (t^{4})^{3}

= (x y^{2} z^{4})^{3} - (t^{4})^{3}

Wende Formel zur Differenz von dritten Potenzen an:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(x y^{2} z^{4})^{3} - (t^{4})^{3} = (x y^{2} z^{4} - t^{4}) ((x y^{2} z^{4})^{2} + x y^{2} z^{4} t^{4} + t^{8})

= (x y^{2} z^{4} - t^{4}) ((x y^{2} z^{4})^{2} + t^{4} x y^{2} z^{4} + t^{8})

Vereinfache

(x y^{2} z^{4} - t^{4}) ((x y^{2} z^{4})^{2} + x y^{2} z^{4} t^{4} + t^{8}) : (x y^{2} z^{4} - t^{4}) (x^{2} y^{4} z^{8} + x y^{2} z^{4} t^{4} + t^{8})

= (x y^{2} z^{4} - t^{4}) (x^{2} y^{4} z^{8} + x y^{2} z^{4} t^{4} + t^{8})

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