Expand Using the Binomial Theorem (r-s)^5

${(r - s)}^{5}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(r)}^{5} \cdot {(- s)}^{0} + 5 \cdot {(r)}^{4} \cdot {(- s)}^{1} + \dots + 1 \cdot {(r)}^{0} \cdot {(- s)}^{5}$

Step 4

Simplify the polynomial result.

$r^{5} - 5 r^{4} s + 10 r^{3} s^{2} - 10 r^{2} s^{3} + 5 r s^{4} - s^{5}$

Expand Using the Binomial Theorem (r-s)^5

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