What is expand (x+iy)^5 ?

The solution to expand (x+iy)^5 is (x^5-10x^3y^2+5xy^4)+(5x^4y-10x^2y^3+y^5)i

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expand (x+iy)^5

expand

(x + i y)^{5}

(x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

Solution steps

(x + i y)^{5}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = i y

= i = 0 \sum 5 (i 5) x^{(5 - i)} (i y)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (i y)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (i y)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (i y)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (i y)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (i y)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (i y)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (i y)^{0} : x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (i y)^{1} : 5 i x^{4} y

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (i y)^{2} : 10 i^{2} x^{3} y^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (i y)^{3} : 10 i^{3} x^{2} y^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (i y)^{4} : 5 i^{4} x y^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (i y)^{5} : i^{5} y^{5}

= x^{5} + 5 i x^{4} y + 10 i^{2} x^{3} y^{2} + 10 i^{3} x^{2} y^{3} + 5 i^{4} x y^{4} + i^{5} y^{5}

10 i^{2} x^{3} y^{2} = - 10 x^{3} y^{2}

10 i^{3} x^{2} y^{3} = - 10 i x^{2} y^{3}

5 i^{4} x y^{4} = 5 x y^{4}

i^{5} = i

= x^{5} + 5 i x^{4} y - 10 x^{3} y^{2} - 10 i x^{2} y^{3} + 5 x y^{4} + i y^{5}

Rewrite in standard complex form:

(x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

= (x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

(5 cos (t) - 5 cos (5 t))^{2}

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

(x - 5) (x^{2} - 9 + 4)

e^{x} (sin (y) + 3 y) d x - (3 x - e^{x} (sin (y)) d y = 0)

(2 - x - 5)^{2}

What is expand (x+iy)^5 ?
The solution to expand (x+iy)^5 is (x^5-10x^3y^2+5xy^4)+(5x^4y-10x^2y^3+y^5)i