faktorisieren x^5y^5+3125

Lösung

faktorisieren

x^{5} y^{5} + 3125

(x y + 5) (x^{4} y^{4} - 5 x^{3} y^{3} + 25 x^{2} y^{2} - 125 x y + 625)

Schritte zur Lösung

x^{5} y^{5} + 3125

Schreibe

x^{5} y^{5}

um:

(x y)^{5}

= (x y)^{5} + 3125

Wende Radikal Regel an:

a = (a)^{2}

3125 = (53125)^{5}

= (x y)^{5} + (53125)^{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})

n is odd

(x y)^{5} + (53125)^{5} = (x y + 5) ((x y)^{4} - 5 (x y)^{3} + 5^{2} (x y)^{2} - 5^{3} x y + 5^{4})

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

Fasse zusammen

= (x y + 5) ((x y)^{4} - 5 (x y)^{3} + 25 (x y)^{2} - 125 x y + 625)

Vereinfache

(x y + 5) ((x y)^{4} - 5 (x y)^{3} + 25 (x y)^{2} - 125 x y + 625) : (x y + 5) (x^{4} y^{4} - 5 x^{3} y^{3} + 25 x^{2} y^{2} - 125 x y + 625)

= (x y + 5) (x^{4} y^{4} - 5 x^{3} y^{3} + 25 x^{2} y^{2} - 125 x y + 625)

f a c t or a^{2} + 2 a (a + b) + (a + b)^{2}

f a c t or 10 + 221

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

f a c t or (x + Δ)^{(2)} - 4 (x + Δ)

f a c t or 17 d^{9} e^{4} - 51 e^{6}