What is expand (x+1/x)^5 ?

The solution to expand (x+1/x)^5 is x^5+5x^3+10x+(10)/x+5/(x^3)+1/(x^5)

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expand (x+1/x)^5

expand

(x + \frac{1}{x})^{5}

x^{5} + 5 x^{3} + 10 x + \frac{10}{x} + \frac{5}{x ^{3}} + \frac{1}{x ^{5}}

Solution steps

(x + \frac{1}{x})^{5}

Apply binomial theorem:

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

a = x, b = \frac{1}{x}

= i = 0 \sum 5 (i 5) x^{(5 - i)} (\frac{1}{x})^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (\frac{1}{x})^{0} : x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (\frac{1}{x})^{1} : 5 x^{3}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (\frac{1}{x})^{2} : 10 x

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (\frac{1}{x})^{3} : \frac{10}{x}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (\frac{1}{x})^{4} : \frac{5}{x ^{3}}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (\frac{1}{x})^{5} : \frac{1}{x ^{5}}

= x^{5} + 5 x^{3} + 10 x + \frac{10}{x} + \frac{5}{x ^{3}} + \frac{1}{x ^{5}}

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

\frac{1}{( 2 - 3 i ) ^{5}}

(x^{2} + \frac{2}{x})^{9}

(3 x - y) (2 x - 5 y)

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

What is expand (x+1/x)^5 ?
The solution to expand (x+1/x)^5 is x^5+5x^3+10x+(10)/x+5/(x^3)+1/(x^5)