Expand Using the Binomial Theorem (x-2y)^13

${(x - 2 y)}^{13}$

Step 1

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

$\sum_{k = 0}^{13} \frac{13!}{(13 - k)! k!} \cdot {(x)}^{13 - k} \cdot {(- 2 y)}^{k}$

Step 2

Expand the summation.

$\frac{13!}{(13 - 0)! 0!} {(x)}^{13 - 0} \cdot {(- 2 y)}^{0} + \frac{13!}{(13 - 1)! 1!} {(x)}^{13 - 1} \cdot {(- 2 y)}^{1} + \dots + \frac{13!}{(13 - 13)! 13!} {(x)}^{13 - 13} \cdot {(- 2 y)}^{13}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{13} \cdot {(- 2 y)}^{0} + 13 \cdot {(x)}^{12} \cdot {(- 2 y)}^{1} + \dots + 1 \cdot {(x)}^{0} \cdot {(- 2 y)}^{13}$

Step 4

Simplify the polynomial result.

$x^{13} - 26 x^{12} y + 312 x^{11} y^{2} - 2288 x^{10} y^{3} + 11440 x^{9} y^{4} - 41184 x^{8} y^{5} + 109824 x^{7} y^{6} - 219648 x^{6} y^{7} + 329472 x^{5} y^{8} - 366080 x^{4} y^{9} + 292864 x^{3} y^{10} - 159744 x^{2} y^{11} + 53248 x y^{12} - 8192 y^{13}$

Expand Using the Binomial Theorem (x-2y)^13

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