Precalculus Examples

${(4 x + 5)}^{5}$

Step 1

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

$\sum_{k = 0}^{5} \frac{5!}{(5 - k)! k!} \cdot {(4 x)}^{5 - k} \cdot {(5)}^{k}$

Step 2

Expand the summation.

$\frac{5!}{(5 - 0)! 0!} \cdot {(4 x)}^{5 - 0} \cdot {(5)}^{0} + \frac{5!}{(5 - 1)! 1!} \cdot {(4 x)}^{5 - 1} \cdot {(5)}^{1} + \dots + \frac{5!}{(5 - 5)! 5!} \cdot {(4 x)}^{5 - 5} \cdot {(5)}^{5}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 x)}^{5} \cdot {(5)}^{0} + 5 \cdot {(4 x)}^{4} \cdot {(5)}^{1} + \dots + 1 \cdot {(4 x)}^{0} \cdot {(5)}^{5}$

Step 4

Simplify the polynomial result.

$1024 x^{5} + 6400 x^{4} + 16000 x^{3} + 20000 x^{2} + 12500 x + 3125$