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

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

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expand (z+1/z)^5

expand

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

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

Solution steps

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

Apply binomial theorem:

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

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

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

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

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

(x + y) (x^{^{'}} + z) (y + z)

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