Precalculus Examples

${(i - 4)}^{6}$

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

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(i)}^{6 - k} \cdot {(- 4)}^{k}$

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} \cdot {(i)}^{6 - 0} \cdot {(- 4)}^{0} + \frac{6!}{(6 - 1)! 1!} \cdot {(i)}^{6 - 1} \cdot {(- 4)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} \cdot {(i)}^{6 - 6} \cdot {(- 4)}^{6}$

Simplify the exponents for each term of the expansion.

$1 \cdot {(i)}^{6} \cdot {(- 4)}^{0} + 6 \cdot {(i)}^{5} \cdot {(- 4)}^{1} + \dots + 1 \cdot {(i)}^{0} \cdot {(- 4)}^{6}$

Simplify the polynomial result.

$495 - 4888 i$