Expand Using the Binomial Theorem (x+(3+5i))^2

${(x + (3 + 5 i))}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(x)}^{2 - k} \cdot {(3 + 5 i)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} {(x)}^{2 - 0} \cdot {(3 + 5 i)}^{0} + \frac{2!}{(2 - 1)! 1!} {(x)}^{2 - 1} \cdot {(3 + 5 i)}^{1} + \frac{2!}{(2 - 2)! 2!} {(x)}^{2 - 2} \cdot {(3 + 5 i)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{2} \cdot {(3 + 5 i)}^{0} + 2 \cdot {(x)}^{1} \cdot {(3 + 5 i)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Step 4

Simplify each term.

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

${(x)}^{2} \cdot {(3 + 5 i)}^{0} + 2 \cdot {(x)}^{1} \cdot {(3 + 5 i)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Anything raised to $0$ is $1$ .

$x^{2} \cdot 1 + 2 \cdot {(x)}^{1} \cdot {(3 + 5 i)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Multiply $x^{2}$ by $1$ .

$x^{2} + 2 \cdot {(x)}^{1} \cdot {(3 + 5 i)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Simplify.

$x^{2} + 2 \cdot x \cdot {(3 + 5 i)}^{1} + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Simplify.

$x^{2} + 2 x \cdot (3 + 5 i) + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Apply the distributive property.

$x^{2} + 2 x \cdot 3 + 2 x (5 i) + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Multiply $3$ by $2$ .

$x^{2} + 6 x + 2 x (5 i) + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Multiply $5$ by $2$ .

$x^{2} + 6 x + 10 x i + 1 \cdot {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Multiply ${(x)}^{0}$ by $1$ .

$x^{2} + 6 x + 10 x i + {(x)}^{0} \cdot {(3 + 5 i)}^{2}$

Anything raised to $0$ is $1$ .

$x^{2} + 6 x + 10 x i + 1 \cdot {(3 + 5 i)}^{2}$

Multiply ${(3 + 5 i)}^{2}$ by $1$ .

$x^{2} + 6 x + 10 x i + {(3 + 5 i)}^{2}$

Rewrite ${(3 + 5 i)}^{2}$ as $(3 + 5 i) (3 + 5 i)$ .

$x^{2} + 6 x + 10 x i + (3 + 5 i) (3 + 5 i)$

Expand $(3 + 5 i) (3 + 5 i)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 6 x + 10 x i + 3 (3 + 5 i) + 5 i (3 + 5 i)$

Apply the distributive property.

$x^{2} + 6 x + 10 x i + 3 \cdot 3 + 3 (5 i) + 5 i (3 + 5 i)$

Apply the distributive property.

$x^{2} + 6 x + 10 x i + 3 \cdot 3 + 3 (5 i) + 5 i \cdot 3 + 5 i (5 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$x^{2} + 6 x + 10 x i + 9 + 3 (5 i) + 5 i \cdot 3 + 5 i (5 i)$

Multiply $5$ by $3$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 5 i \cdot 3 + 5 i (5 i)$

Multiply $3$ by $5$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 5 i (5 i)$

Multiply $5 i (5 i)$ .

Multiply $5$ by $5$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 i i$

Raise $i$ to the power of $1$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 (i^{1} i)$

Raise $i$ to the power of $1$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 i^{1 + 1}$

Add $1$ and $1$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i + 25 \cdot - 1$

Multiply $25$ by $- 1$ .

$x^{2} + 6 x + 10 x i + 9 + 15 i + 15 i - 25$

Subtract $25$ from $9$ .

$x^{2} + 6 x + 10 x i - 16 + 15 i + 15 i$

Add $15 i$ and $15 i$ .

$x^{2} + 6 x + 10 x i - 16 + 30 i$

Expand Using the Binomial Theorem (x+(3+5i))^2

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