vereinfachen (x+iy)^3+2i(x+iy)^2+5(x+iy)+2

Lösung

vereinfachen

(x + i y)^{3} + 2 i (x + i y)^{2} + 5 (x + i y) + 2

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

Schritte zur Lösung

(x + i y)^{3} + 2 i (x + i y)^{2} + 5 (x + i y) + 2

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

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

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

Schreibe

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

um:

2 x^{2} i - 4 x y - 2 i y^{2}

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

Schreibe

5 (x + i y)

um:

5 x + 5 y i

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

Rewrite in standard complex form:

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

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

s im pl i f y 2 (x + 2)

e x p an d y (x^{2} + y^{2} - 1) d x + x (x^{2} + y^{2} + 1) d y

e x p an d \frac{( x + 2 ) ( 4 x - 1 ) ^{2}}{x ^{3}}

s im pl i f y (x^{2} + 5) (1 - x)

s im pl i f y (x - 9) (3 x^{2} - 12)