Expand Using the Binomial Theorem (1/x+y)^5

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

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{5} \frac{5!}{(5 - k)! k!} \cdot {(\frac{1}{x})}^{5 - k} \cdot {(y)}^{k}$

Expand the summation.

$\frac{5!}{(5 - 0)! 0!} \cdot {(\frac{1}{x})}^{5 - 0} \cdot {(y)}^{0} + \frac{5!}{(5 - 1)! 1!} \cdot {(\frac{1}{x})}^{5 - 1} \cdot {(y)}^{1} + \dots + \frac{5!}{(5 - 5)! 5!} \cdot {(\frac{1}{x})}^{5 - 5} \cdot {(y)}^{5}$

Simplify the exponents for each term of the expansion.

$1 \cdot {(\frac{1}{x})}^{5} \cdot {(y)}^{0} + 5 \cdot {(\frac{1}{x})}^{4} \cdot {(y)}^{1} + \dots + 1 \cdot {(\frac{1}{x})}^{0} \cdot {(y)}^{5}$

Simplify the polynomial result.

$\frac{1}{x^{5}} + \frac{5 y}{x^{4}} + \frac{10 y^{2}}{x^{3}} + \frac{10 y^{3}}{x^{2}} + \frac{5 y^{4}}{x} + y^{5}$

Expand Using the Binomial Theorem (1/x+y)^5

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