Expand Using the Binomial Theorem (x^3+1/x)^10

${(x^{3} + \frac{1}{x})}^{10}$

Step 1

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

$\sum_{k = 0}^{10} \frac{10!}{(10 - k)! k!} \cdot {(x^{3})}^{10 - k} \cdot {(\frac{1}{x})}^{k}$

Step 2

Expand the summation.

$\frac{10!}{(10 - 0)! 0!} \cdot {(x^{3})}^{10 - 0} \cdot {(\frac{1}{x})}^{0} + \frac{10!}{(10 - 1)! 1!} \cdot {(x^{3})}^{10 - 1} \cdot {(\frac{1}{x})}^{1} + \dots + \frac{10!}{(10 - 10)! 10!} \cdot {(x^{3})}^{10 - 10} \cdot {(\frac{1}{x})}^{10}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x^{3})}^{10} \cdot {(\frac{1}{x})}^{0} + 10 \cdot {(x^{3})}^{9} \cdot {(\frac{1}{x})}^{1} + \dots + 1 \cdot {(x^{3})}^{0} \cdot {(\frac{1}{x})}^{10}$

Step 4

Simplify the polynomial result.

$x^{30} + 10 x^{26} + 45 x^{22} + 120 x^{18} + 210 x^{14} + 252 x^{10} + 210 x^{6} + 120 x^{2} + \frac{45}{x^{2}} + \frac{10}{x^{6}} + \frac{1}{x^{10}}$

Expand Using the Binomial Theorem (x^3+1/x)^10

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