Precalculus Examples

${(6 a + b)}^{6}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(6 a)}^{6} \cdot {(b)}^{0} + 6 \cdot {(6 a)}^{5} \cdot {(b)}^{1} + \dots + 1 \cdot {(6 a)}^{0} \cdot {(b)}^{6}$

Step 4

Simplify the polynomial result.

$46656 a^{6} + 46656 a^{5} b + 19440 a^{4} b^{2} + 4320 a^{3} b^{3} + 540 a^{2} b^{4} + 36 a b^{5} + b^{6}$