Precalculus Examples

${(x - 3)}^{11}$

Step 1

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

$\sum_{k = 0}^{11} \frac{11!}{(11 - k)! k!} \cdot {(x)}^{11 - k} \cdot {(- 3)}^{k}$

Step 2

Expand the summation.

$\frac{11!}{(11 - 0)! 0!} {(x)}^{11 - 0} \cdot {(- 3)}^{0} + \frac{11!}{(11 - 1)! 1!} {(x)}^{11 - 1} \cdot {(- 3)}^{1} + \dots + \frac{11!}{(11 - 11)! 11!} {(x)}^{11 - 11} \cdot {(- 3)}^{11}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{11} \cdot {(- 3)}^{0} + 11 \cdot {(x)}^{10} \cdot {(- 3)}^{1} + \dots + 1 \cdot {(x)}^{0} \cdot {(- 3)}^{11}$

Step 4

Simplify the polynomial result.

$x^{11} - 33 x^{10} + 495 x^{9} - 4455 x^{8} + 26730 x^{7} - 112266 x^{6} + 336798 x^{5} - 721710 x^{4} + 1082565 x^{3} - 1082565 x^{2} + 649539 x - 177147$