Expand Using the Binomial Theorem (2a+b)^2

${(2 a + b)}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(2 a)}^{2 - k} \cdot {(b)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(2 a)}^{2 - 0} \cdot {(b)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(2 a)}^{2 - 1} \cdot {(b)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(2 a)}^{2 - 2} \cdot {(b)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 a)}^{2} \cdot {(b)}^{0} + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Step 4

Simplify each term.

Multiply ${(2 a)}^{2}$ by $1$ .

${(2 a)}^{2} \cdot {(b)}^{0} + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Apply the product rule to $2 a$ .

$2^{2} a^{2} \cdot {(b)}^{0} + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Raise $2$ to the power of $2$ .

$4 a^{2} \cdot {(b)}^{0} + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Anything raised to $0$ is $1$ .

$4 a^{2} \cdot 1 + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Multiply $4$ by $1$ .

$4 a^{2} + 2 \cdot {(2 a)}^{1} \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Simplify.

$4 a^{2} + 2 \cdot (2 a) \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Multiply $2$ by $2$ .

$4 a^{2} + 4 a \cdot {(b)}^{1} + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Simplify.

$4 a^{2} + 4 a \cdot b + 1 \cdot {(2 a)}^{0} \cdot {(b)}^{2}$

Multiply ${(2 a)}^{0}$ by $1$ .

$4 a^{2} + 4 a b + {(2 a)}^{0} \cdot {(b)}^{2}$

Apply the product rule to $2 a$ .

$4 a^{2} + 4 a b + 2^{0} a^{0} \cdot {(b)}^{2}$

Anything raised to $0$ is $1$ .

$4 a^{2} + 4 a b + 1 a^{0} \cdot {(b)}^{2}$

Multiply $a^{0}$ by $1$ .

$4 a^{2} + 4 a b + a^{0} \cdot {(b)}^{2}$

Anything raised to $0$ is $1$ .

$4 a^{2} + 4 a b + 1 \cdot {(b)}^{2}$

Multiply ${(b)}^{2}$ by $1$ .

$4 a^{2} + 4 a b + b^{2}$

Expand Using the Binomial Theorem (2a+b)^2

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