Precalculus Examples

${(w - s)}^{6}$

Step 1

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

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(w)}^{6 - k} \cdot {(- s)}^{k}$

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} {(w)}^{6 - 0} \cdot {(- s)}^{0} + \frac{6!}{(6 - 1)! 1!} {(w)}^{6 - 1} \cdot {(- s)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} {(w)}^{6 - 6} \cdot {(- s)}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(w)}^{6} \cdot {(- s)}^{0} + 6 \cdot {(w)}^{5} \cdot {(- s)}^{1} + \dots + 1 \cdot {(w)}^{0} \cdot {(- s)}^{6}$

Step 4

Simplify the polynomial result.

$w^{6} - 6 w^{5} s + 15 w^{4} s^{2} - 20 w^{3} s^{3} + 15 w^{2} s^{4} - 6 w s^{5} + s^{6}$