Expand Using the Binomial Theorem (a+b)^9

${(a + b)}^{9}$

Step 1

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

$\sum_{k = 0}^{9} \frac{9!}{(9 - k)! k!} \cdot {(a)}^{9 - k} \cdot {(b)}^{k}$

Step 2

Expand the summation.

$\frac{9!}{(9 - 0)! 0!} {(a)}^{9 - 0} \cdot {(b)}^{0} + \frac{9!}{(9 - 1)! 1!} {(a)}^{9 - 1} \cdot {(b)}^{1} + \dots + \frac{9!}{(9 - 9)! 9!} {(a)}^{9 - 9} \cdot {(b)}^{9}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(a)}^{9} \cdot {(b)}^{0} + 9 \cdot {(a)}^{8} \cdot {(b)}^{1} + \dots + 1 \cdot {(a)}^{0} \cdot {(b)}^{9}$

Step 4

Simplify the polynomial result.

$a^{9} + 9 a^{8} b + 36 a^{7} b^{2} + 84 a^{6} b^{3} + 126 a^{5} b^{4} + 126 a^{4} b^{5} + 84 a^{3} b^{6} + 36 a^{2} b^{7} + 9 a b^{8} + b^{9}$

Expand Using the Binomial Theorem (a+b)^9

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