Expand Using the Binomial Theorem (2A+B^2)^5

${(2 A + B^{2})}^{5}$

Step 1

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

$\sum_{k = 0}^{5} \frac{5!}{(5 - k)! k!} \cdot {(2 A)}^{5 - k} \cdot {(B^{2})}^{k}$

Step 2

Expand the summation.

$\frac{5!}{(5 - 0)! 0!} \cdot {(2 A)}^{5 - 0} \cdot {(B^{2})}^{0} + \frac{5!}{(5 - 1)! 1!} \cdot {(2 A)}^{5 - 1} \cdot {(B^{2})}^{1} + \dots + \frac{5!}{(5 - 5)! 5!} \cdot {(2 A)}^{5 - 5} \cdot {(B^{2})}^{5}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 A)}^{5} \cdot {(B^{2})}^{0} + 5 \cdot {(2 A)}^{4} \cdot {(B^{2})}^{1} + \dots + 1 \cdot {(2 A)}^{0} \cdot {(B^{2})}^{5}$

Step 4

Simplify the polynomial result.

$32 A^{5} + 80 A^{4} B^{2} + 80 A^{3} B^{4} + 40 A^{2} B^{6} + 10 A B^{8} + B^{10}$

Expand Using the Binomial Theorem (2A+B^2)^5

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