Precalculus Examples

${(x - 3 y)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(x)}^{7 - k} \cdot {(- 3 y)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} {(x)}^{7 - 0} \cdot {(- 3 y)}^{0} + \frac{7!}{(7 - 1)! 1!} {(x)}^{7 - 1} \cdot {(- 3 y)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} {(x)}^{7 - 7} \cdot {(- 3 y)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{7} \cdot {(- 3 y)}^{0} + 7 \cdot {(x)}^{6} \cdot {(- 3 y)}^{1} + \dots + 1 \cdot {(x)}^{0} \cdot {(- 3 y)}^{7}$

Step 4

Simplify the polynomial result.

$x^{7} - 21 x^{6} y + 189 x^{5} y^{2} - 945 x^{4} y^{3} + 2835 x^{3} y^{4} - 5103 x^{2} y^{5} + 5103 x y^{6} - 2187 y^{7}$