Expand Using the Binomial Theorem (5z+2)^2

${(5 z + 2)}^{2}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5 z)}^{2} \cdot {(2)}^{0} + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Step 4

Simplify each term.

Multiply ${(5 z)}^{2}$ by $1$ .

${(5 z)}^{2} \cdot {(2)}^{0} + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Apply the product rule to $5 z$ .

$5^{2} z^{2} \cdot {(2)}^{0} + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Raise $5$ to the power of $2$ .

$25 z^{2} \cdot {(2)}^{0} + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Anything raised to $0$ is $1$ .

$25 z^{2} \cdot 1 + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Multiply $25$ by $1$ .

$25 z^{2} + 2 \cdot {(5 z)}^{1} \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Simplify.

$25 z^{2} + 2 \cdot (5 z) \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Multiply $5$ by $2$ .

$25 z^{2} + 10 z \cdot {(2)}^{1} + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Evaluate the exponent.

$25 z^{2} + 10 z \cdot 2 + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Multiply $2$ by $10$ .

$25 z^{2} + 20 z + 1 \cdot {(5 z)}^{0} \cdot {(2)}^{2}$

Multiply ${(5 z)}^{0}$ by $1$ .

$25 z^{2} + 20 z + {(5 z)}^{0} \cdot {(2)}^{2}$

Apply the product rule to $5 z$ .

$25 z^{2} + 20 z + 5^{0} z^{0} \cdot {(2)}^{2}$

Anything raised to $0$ is $1$ .

$25 z^{2} + 20 z + 1 z^{0} \cdot {(2)}^{2}$

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

$25 z^{2} + 20 z + z^{0} \cdot {(2)}^{2}$

Anything raised to $0$ is $1$ .

$25 z^{2} + 20 z + 1 \cdot {(2)}^{2}$

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

$25 z^{2} + 20 z + {(2)}^{2}$

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

$25 z^{2} + 20 z + 4$

Expand Using the Binomial Theorem (5z+2)^2

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