Expand Using the Binomial Theorem (p+q)^7

${(p + q)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(p)}^{7 - k} \cdot {(q)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} {(p)}^{7 - 0} \cdot {(q)}^{0} + \frac{7!}{(7 - 1)! 1!} {(p)}^{7 - 1} \cdot {(q)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} {(p)}^{7 - 7} \cdot {(q)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(p)}^{7} \cdot {(q)}^{0} + 7 \cdot {(p)}^{6} \cdot {(q)}^{1} + \dots + 1 \cdot {(p)}^{0} \cdot {(q)}^{7}$

Step 4

Simplify the polynomial result.

$p^{7} + 7 p^{6} q + 21 p^{5} q^{2} + 35 p^{4} q^{3} + 35 p^{3} q^{4} + 21 p^{2} q^{5} + 7 p q^{6} + q^{7}$

Expand Using the Binomial Theorem (p+q)^7

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