Simplify (x-3)(x-5i)(x+5i)

$(x - 3) (x - 5 i) (x + 5 i)$

Step 1

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

Apply the distributive property.

$(x (x - 5 i) - 3 (x - 5 i)) (x + 5 i)$

Apply the distributive property.

$(x \cdot x + x (- 5 i) - 3 (x - 5 i)) (x + 5 i)$

Apply the distributive property.

$(x \cdot x + x (- 5 i) - 3 x - 3 (- 5 i)) (x + 5 i)$

Step 2

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $- 3$ .

$(x^{2} + x (- 5 i) - 3 x + 15 i) (x + 5 i)$

Step 3

Expand $(x^{2} + x (- 5 i) - 3 x + 15 i) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Step 4

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

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

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

$x^{2} x^{1} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

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

$x^{2 + 1} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Add $2$ and $1$ .

$x^{3} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Move $5$ to the left of $x^{2}$ .

$x^{3} + 5 \cdot x^{2} i + x (- 5 i) x + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{3} + 5 x^{2} i + x \cdot x (- 5 i) + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $x$ by $x$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 5 i) (5 i) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $x (- 5 i) (5 i)$ .

Multiply $5$ by $- 5$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i) i - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 (i^{1} i)) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 (i^{1} i^{1})) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i^{1 + 1}) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Add $1$ and $1$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i^{2}) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 \cdot - 1) - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $- 25$ by $- 1$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x \cdot 25 - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Move $25$ to the left of $x$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 \cdot x - 3 x \cdot x - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 (x \cdot x) - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $x$ by $x$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 3 x (5 i) + 15 i x + 15 i (5 i)$

Multiply $5$ by $- 3$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 15 i (5 i)$

Multiply $15 i (5 i)$ .

Multiply $5$ by $15$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 i i$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 (i^{1} i)$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 (i^{1} i^{1})$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 i^{1 + 1}$

Add $1$ and $1$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 i^{2}$

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

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x + 75 \cdot - 1$

Multiply $75$ by $- 1$ .

$x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x - 75$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - 3 x^{2} - 15 x i + 15 i x - 75$ .

Reorder the factors in the terms $5 x^{2} i$ and $x^{2} (- 5 i)$ .

$x^{3} + 5 i x^{2} - 5 i x^{2} + 25 x - 3 x^{2} - 15 x i + 15 i x - 75$

Subtract $5 i x^{2}$ from $5 i x^{2}$ .

$x^{3} + 0 + 25 x - 3 x^{2} - 15 x i + 15 i x - 75$

Add $x^{3}$ and $0$ .

$x^{3} + 25 x - 3 x^{2} - 15 x i + 15 i x - 75$

Reorder the factors in the terms $- 15 x i$ and $15 i x$ .

$x^{3} + 25 x - 3 x^{2} - 15 i x + 15 i x - 75$

Add $- 15 i x$ and $15 i x$ .

$x^{3} + 25 x - 3 x^{2} + 0 - 75$

Add $x^{3} + 25 x - 3 x^{2}$ and $0$ .

$x^{3} + 25 x - 3 x^{2} - 75$

Move $25 x$ .

$x^{3} - 3 x^{2} + 25 x - 75$

Simplify (x-3)(x-5i)(x+5i)

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