Expand Using the Binomial Theorem (2u-v)^6

${(2 u - v)}^{6}$

Step 1

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

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(2 u)}^{6 - k} \cdot {(- v)}^{k}$

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} \cdot {(2 u)}^{6 - 0} \cdot {(- v)}^{0} + \frac{6!}{(6 - 1)! 1!} \cdot {(2 u)}^{6 - 1} \cdot {(- v)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} \cdot {(2 u)}^{6 - 6} \cdot {(- v)}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 u)}^{6} \cdot {(- v)}^{0} + 6 \cdot {(2 u)}^{5} \cdot {(- v)}^{1} + \dots + 1 \cdot {(2 u)}^{0} \cdot {(- v)}^{6}$

Step 4

Simplify the polynomial result.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + v^{6}$

Expand Using the Binomial Theorem (2u-v)^6

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