erweitern ((x-y)^2+2)^3

Lösung

erweitern

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

x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6} + 6 x^{4} - 24 x^{3} y + 36 x^{2} y^{2} - 24 x y^{3} + 6 y^{4} + 12 x^{2} - 24 x y + 12 y^{2} + 8

Schritte zur Lösung

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

Wende Formel für perfekte dritte Potenzen an:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

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

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

Vereinfache

((x - y)^{2})^{3} + 3 ((x - y)^{2})^{2} \cdot 2 + 3 (x - y)^{2} \cdot 2^{2} + 2^{3} : (x - y)^{6} + 6 (x - y)^{4} + 12 (x - y)^{2} + 8

= (x - y)^{6} + 6 (x - y)^{4} + 12 (x - y)^{2} + 8

Schreibe

(x - y)^{6}

um:

x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6}

= x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6} + 6 (x - y)^{4} + 12 (x - y)^{2} + 8

Schreibe

6 (x - y)^{4}

um:

6 x^{4} - 24 x^{3} y + 36 x^{2} y^{2} - 24 x y^{3} + 6 y^{4}

= x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6} + 6 x^{4} - 24 x^{3} y + 36 x^{2} y^{2} - 24 x y^{3} + 6 y^{4} + 12 (x - y)^{2} + 8

Schreibe

12 (x - y)^{2}

um:

12 x^{2} - 24 x y + 12 y^{2}

= x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6} + 6 x^{4} - 24 x^{3} y + 36 x^{2} y^{2} - 24 x y^{3} + 6 y^{4} + 12 x^{2} - 24 x y + 12 y^{2} + 8

s im pl i f y \frac{3}{x - 3} + \frac{x}{x + 3}

s im pl i f y (- 2 x^{4} y) (x y^{4}) (3 x y)

s im pl i f y ∣ 6 - 2 ∣

f a c t or 1 - x - 2 x^{2}

s im pl i f y - 6 y + y - 7 x