Finite Math Examples

${(2 c - d^{2})}^{9}$

Step 1

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

$\sum_{k = 0}^{9} \frac{9!}{(9 - k)! k!} \cdot {(2 c)}^{9 - k} \cdot {(- d^{2})}^{k}$

Step 2

Expand the summation.

$\frac{9!}{(9 - 0)! 0!} \cdot {(2 c)}^{9 - 0} \cdot {(- d^{2})}^{0} + \frac{9!}{(9 - 1)! 1!} \cdot {(2 c)}^{9 - 1} \cdot {(- d^{2})}^{1} + \dots + \frac{9!}{(9 - 9)! 9!} \cdot {(2 c)}^{9 - 9} \cdot {(- d^{2})}^{9}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 c)}^{9} \cdot {(- d^{2})}^{0} + 9 \cdot {(2 c)}^{8} \cdot {(- d^{2})}^{1} + \dots + 1 \cdot {(2 c)}^{0} \cdot {(- d^{2})}^{9}$

Step 4

Simplify the polynomial result.

$512 c^{9} - 2304 c^{8} d^{2} + 4608 c^{7} d^{4} - 5376 c^{6} d^{6} + 4032 c^{5} d^{8} - 2016 c^{4} d^{10} + 672 c^{3} d^{12} - 144 c^{2} d^{14} + 18 c d^{16} - d^{18}$