What is expand (1/x+3x^2)^5 ?

The solution to expand (1/x+3x^2)^5 is 1/(x^5)+(15)/(x^2)+90x+270x^4+405x^7+243x^{10}

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expand (1/x+3x^2)^5

expand

(\frac{1}{x} + 3 x^{2})^{5}

\frac{1}{x ^{5}} + \frac{15}{x ^{2}} + 90 x + 270 x^{4} + 405 x^{7} + 243 x^{10}

Solution steps

(\frac{1}{x} + 3 x^{2})^{5}

Apply binomial theorem:

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

a = \frac{1}{x}, b = 3 x^{2}

= i = 0 \sum 5 (i 5) (\frac{1}{x})^{(5 - i)} (3 x^{2})^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{x})^{5} (3 x^{2})^{0} : \frac{1}{x ^{5}}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{x})^{4} (3 x^{2})^{1} : \frac{15}{x ^{2}}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{x})^{3} (3 x^{2})^{2} : 90 x

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (\frac{1}{x})^{2} (3 x^{2})^{3} : 270 x^{4}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{x})^{1} (3 x^{2})^{4} : 405 x^{7}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{x})^{0} (3 x^{2})^{5} : 243 x^{10}

= \frac{1}{x ^{5}} + \frac{15}{x ^{2}} + 90 x + 270 x^{4} + 405 x^{7} + 243 x^{10}

- \frac{( k + 1 ) ( 25 k + 41 )}{2 ( - 15 k + 1 )}

(\frac{2 n}{7} - 2 x^{2}) \cdot 3 y^{4}

(x^{3} - \frac{1}{x ^{2}})^{3}

\frac{3}{2 ( x - 2 ) ^{2}}

, (2 t^{2} - 1) 2 d t

What is expand (1/x+3x^2)^5 ?
The solution to expand (1/x+3x^2)^5 is 1/(x^5)+(15)/(x^2)+90x+270x^4+405x^7+243x^{10}