Expand Using the Binomial Theorem (p-2q)^6

${(p - 2 q)}^{6}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify the polynomial result.

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

Expand Using the Binomial Theorem (p-2q)^6

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