Expand Using the Binomial Theorem (x^2+1/x)^12

${(x^{2} + \frac{1}{x})}^{12}$

Step 1

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

$\sum_{k = 0}^{12} \frac{12!}{(12 - k)! k!} \cdot {(x^{2})}^{12 - k} \cdot {(\frac{1}{x})}^{k}$

Step 2

Expand the summation.

$\frac{12!}{(12 - 0)! 0!} \cdot {(x^{2})}^{12 - 0} \cdot {(\frac{1}{x})}^{0} + \frac{12!}{(12 - 1)! 1!} \cdot {(x^{2})}^{12 - 1} \cdot {(\frac{1}{x})}^{1} + \dots + \frac{12!}{(12 - 12)! 12!} \cdot {(x^{2})}^{12 - 12} \cdot {(\frac{1}{x})}^{12}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x^{2})}^{12} \cdot {(\frac{1}{x})}^{0} + 12 \cdot {(x^{2})}^{11} \cdot {(\frac{1}{x})}^{1} + \dots + 1 \cdot {(x^{2})}^{0} \cdot {(\frac{1}{x})}^{12}$

Step 4

Simplify the polynomial result.

$x^{24} + 12 x^{21} + 66 x^{18} + 220 x^{15} + 495 x^{12} + 792 x^{9} + 924 x^{6} + 792 x^{3} + 495 + \frac{220}{x^{3}} + \frac{66}{x^{6}} + \frac{12}{x^{9}} + \frac{1}{x^{12}}$

Expand Using the Binomial Theorem (x^2+1/x)^12

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