faktorisieren 27x^3-(x-y)^9

Lösung

faktorisieren

27 x^{3} - (x - y)^{9}

(3 x - x^{3} + 3 x^{2} y - 3 x y^{2} + y^{3}) (9 x^{2} + 3 x (x - y)^{3} + (x - y)^{6})

Schritte zur Lösung

27 x^{3} - (x - y)^{9}

Schreibe

27 x^{3} - (x - y)^{9}

um:

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

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

Wende Formel zur Differenz von dritten Potenzen an:

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

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

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

Fasse zusammen

= (3 x - (x - y)^{3}) (9 x^{2} + 3 x (x - y)^{3} + (x - y)^{6})

Multipliziere aus

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

= (- x^{3} + 3 x^{2} y + 3 x - 3 x y^{2} + y^{3}) (9 x^{2} + 3 x (x - y)^{3} + (x - y)^{6})

f a c t or 9 72 + 32

f a c t or 3^{2} - 12

s im pl i f y [5.1]

f a c t or 4 p + 8 r - pq - 2 q r

f (x) = (arcsin (x))^{3}