simplify (x+yi)^3

Solution

simplify

(x + y i)^{3}

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

Solution steps

(x + y i)^{3}

Apply Perfect Cube Formula:

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

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

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

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

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

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

Rewrite in standard complex form:

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

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

(sec (x) - tan (x))^{2}

\frac{- x ^{2} + 1}{( x ^{2} + 1 ) ^{2}}

x (x + 1) (x + 2) (x + 3) + 1

(x - 3) (x + 3)

(x - 2) (x + 2)