faktorisieren 32x^5-243y^{10}z^5

Lösung

faktorisieren

32 x^{5} - 243 y^{10} z^{5}

(2 x - 3 y^{2} z) (16 x^{4} + 24 x^{3} y^{2} z + 36 x^{2} y^{4} z^{2} + 54 x y^{6} z^{3} + 81 y^{8} z^{4})

Schritte zur Lösung

32 x^{5} - 243 y^{10} z^{5}

Schreibe

32 x^{5}

um:

(2 x)^{5}

Schreibe

243 y^{10} z^{5}

um:

(3 y^{2} z)^{5}

= (2 x)^{5} - (3 y^{2} z)^{5}

Wende Regel zum Faktorisieren an:

x^{n} - y^{n} = (x - y) (x^{n - 1} + x^{n - 2} y + \dots + x y^{n - 2} y^{n - 1})

(2 x)^{5} - (3 y^{2} z)^{5} = (2 x - 3 y^{2} z) ((2 x)^{4} + (2 x)^{3} 3 y^{2} z + (2 x)^{2} (3 y^{2} z)^{2} + 2 x (3 y^{2} z)^{3} + (3 y^{2} z)^{4})

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

Vereinfache

(2 x - 3 y^{2} z) ((2 x)^{4} + (2 x)^{3} \cdot 3 y^{2} z + (2 x)^{2} (3 y^{2} z)^{2} + 2 x (3 y^{2} z)^{3} + (3 y^{2} z)^{4}) : (2 x - 3 y^{2} z) (16 x^{4} + 24 x^{3} y^{2} z + 36 x^{2} y^{4} z^{2} + 54 x y^{6} z^{3} + 81 y^{8} z^{4})

= (2 x - 3 y^{2} z) (16 x^{4} + 24 x^{3} y^{2} z + 36 x^{2} y^{4} z^{2} + 54 x y^{6} z^{3} + 81 y^{8} z^{4})

s im pl i f y - 12

f a c t or e^{- \infty} - e^{\infty}

f a c t or 2 - 5 s + 6

f a c t or a e^{x} sin (x) + b e^{x} cos (x)

f a c t or - 2 (x + 3)^{2} + 5