Precalculus Examples

$(x - 4) (x + 2 i) (x - 2 i)$

Step 1

Expand $(x - 4) (x + 2 i)$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$(x (x + 2 i) - 4 (x + 2 i)) (x - 2 i)$

Step 1.2

Apply the distributive property.

$(x \cdot x + x (2 i) - 4 (x + 2 i)) (x - 2 i)$

Step 1.3

Apply the distributive property.

$(x \cdot x + x (2 i) - 4 x - 4 (2 i)) (x - 2 i)$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Multiply $x$ by $x$ .

$(x^{2} + x (2 i) - 4 x - 4 (2 i)) (x - 2 i)$

Step 2.2

Move $2$ to the left of $x$ .

$(x^{2} + 2 \cdot x i - 4 x - 4 (2 i)) (x - 2 i)$

Step 2.3

Multiply $2$ by $- 4$ .

$(x^{2} + 2 x i - 4 x - 8 i) (x - 2 i)$

Step 3

Expand $(x^{2} + 2 x i - 4 x - 8 i) (x - 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x + x^{2} (- 2 i) + 2 x i x + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

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

Tap for more steps...

Step 4.1.1.1.1

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

$x^{2} x^{1} + x^{2} (- 2 i) + 2 x i x + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.1.1.2

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

$x^{2 + 1} + x^{2} (- 2 i) + 2 x i x + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} (- 2 i) + 2 x i x + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Move $x$ .

$x^{3} + x^{2} (- 2 i) + 2 (x \cdot x) i + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.2.2

Multiply $x$ by $x$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 2 x i (- 2 i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.3

Multiply $2 x i (- 2 i)$ .

Tap for more steps...

Step 4.1.3.1

Multiply $- 2$ by $2$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x i i - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.3.2

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x (i^{1} i) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.3.3

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x (i^{1} i^{1}) - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.3.4

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x i^{1 + 1} - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.3.5

Add $1$ and $1$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x i^{2} - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.4

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i - 4 x \cdot - 1 - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.5

Multiply $- 1$ by $- 4$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x \cdot x - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.6

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

Tap for more steps...

Step 4.1.6.1

Move $x$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 (x \cdot x) - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.6.2

Multiply $x$ by $x$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} - 4 x (- 2 i) - 8 i x - 8 i (- 2 i)$

Step 4.1.7

Multiply $- 2$ by $- 4$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x - 8 i (- 2 i)$

Step 4.1.8

Multiply $- 8 i (- 2 i)$ .

Tap for more steps...

Step 4.1.8.1

Multiply $- 2$ by $- 8$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 i i$

Step 4.1.8.2

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 (i^{1} i)$

Step 4.1.8.3

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 (i^{1} i^{1})$

Step 4.1.8.4

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 i^{1 + 1}$

Step 4.1.8.5

Add $1$ and $1$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 i^{2}$

Step 4.1.9

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

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x + 16 \cdot - 1$

Step 4.1.10

Multiply $16$ by $- 1$ .

$x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x - 16$

Step 4.2

Simplify by adding terms.

Tap for more steps...

Step 4.2.1

Combine the opposite terms in $x^{3} + x^{2} (- 2 i) + 2 x^{2} i + 4 x - 4 x^{2} + 8 x i - 8 i x - 16$ .

Tap for more steps...

Step 4.2.1.1

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

$x^{3} - 2 i x^{2} + 2 i x^{2} + 4 x - 4 x^{2} + 8 x i - 8 i x - 16$

Step 4.2.1.2

Add $- 2 i x^{2}$ and $2 i x^{2}$ .

$x^{3} + 0 + 4 x - 4 x^{2} + 8 x i - 8 i x - 16$

Step 4.2.1.3

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

$x^{3} + 4 x - 4 x^{2} + 8 x i - 8 i x - 16$

Step 4.2.1.4

Reorder the factors in the terms $8 x i$ and $- 8 i x$ .

$x^{3} + 4 x - 4 x^{2} + 8 i x - 8 i x - 16$

Step 4.2.1.5

Subtract $8 i x$ from $8 i x$ .

$x^{3} + 4 x - 4 x^{2} + 0 - 16$

Step 4.2.1.6

Add $x^{3} + 4 x - 4 x^{2}$ and $0$ .

$x^{3} + 4 x - 4 x^{2} - 16$

Step 4.2.2

Move $4 x$ .

$x^{3} - 4 x^{2} + 4 x - 16$