Expand Using the Binomial Theorem (p+q)^5

${(p + q)}^{5}$

Step 1

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

$\sum_{k = 0}^{5} \frac{5!}{(5 - k)! k!} \cdot {(p)}^{5 - k} \cdot {(q)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(p)}^{5} \cdot {(q)}^{0} + 5 \cdot {(p)}^{4} \cdot {(q)}^{1} + \dots + 1 \cdot {(p)}^{0} \cdot {(q)}^{5}$

Step 4

Simplify the polynomial result.

$p^{5} + 5 p^{4} q + 10 p^{3} q^{2} + 10 p^{2} q^{3} + 5 p q^{4} + q^{5}$

Expand Using the Binomial Theorem (p+q)^5

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