Precalculus Examples

$(x - 2) (x - (3 + i)) (x - (3 - i))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Apply the distributive property.

$(x - 2) (x - 1 \cdot 3 - i) (x - (3 - i))$

Step 1.2

Multiply $- 1$ by $3$ .

$(x - 2) (x - 3 - i) (x - (3 - i))$

Step 2

Expand $(x - 2) (x - 3 - i)$ by multiplying each term in the first expression by each term in the second expression.

$(x \cdot x + x \cdot - 3 + x (- i) - 2 x - 2 \cdot - 3 - 2 (- i)) (x - (3 - i))$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 3$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 2$ by $- 3$ .

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

Step 3.1.4

Multiply $- 1$ by $- 2$ .

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

Step 3.2

Subtract $2 x$ from $- 3 x$ .

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

Step 3.3

Simplify each term.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $- 1$ by $3$ .

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

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$(x^{2} + x (- i) - 5 x + 6 + 2 i) (x - 3 + 1 i)$

Step 3.3.4

Multiply $i$ by $1$ .

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

Step 4

Expand $(x^{2} + x (- i) - 5 x + 6 + 2 i) (x - 3 + i)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

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

Tap for more steps...

Step 5.1.1.1

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

Tap for more steps...

Step 5.1.1.1.1

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

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

Step 5.1.1.1.2

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

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

Step 5.1.1.2

Add $2$ and $1$ .

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

Step 5.1.2

Move $- 3$ to the left of $x^{2}$ .

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

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

Move $x$ .

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

Step 5.1.3.2

Multiply $x$ by $x$ .

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

Step 5.1.4

Multiply $- 3$ by $- 1$ .

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

Step 5.1.5

Move $3$ to the left of $x$ .

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

Step 5.1.6

Multiply $x (- i) i$ .

Tap for more steps...

Step 5.1.6.1

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

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

Step 5.1.6.2

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

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

Step 5.1.6.3

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

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

Step 5.1.6.4

Add $1$ and $1$ .

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

Step 5.1.7

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

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

Step 5.1.8

Multiply $- 1$ by $- 1$ .

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

Step 5.1.9

Multiply $x$ by $1$ .

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

Step 5.1.10

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

Tap for more steps...

Step 5.1.10.1

Move $x$ .

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

Step 5.1.10.2

Multiply $x$ by $x$ .

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

Step 5.1.11

Multiply $- 3$ by $- 5$ .

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

Step 5.1.12

Multiply $6$ by $- 3$ .

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x + 2 i \cdot - 3 + 2 i i$

Step 5.1.13

Multiply $- 3$ by $2$ .

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 i i$

Step 5.1.14

Multiply $2 i i$ .

Tap for more steps...

Step 5.1.14.1

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

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 (i^{1} i)$

Step 5.1.14.2

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

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 (i^{1} i^{1})$

Step 5.1.14.3

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

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 i^{1 + 1}$

Step 5.1.14.4

Add $1$ and $1$ .

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 i^{2}$

Step 5.1.15

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

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i + 2 \cdot - 1$

Step 5.1.16

Multiply $2$ by $- 1$ .

$x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i - 2$

Step 5.2

Simplify by adding terms.

Tap for more steps...

Step 5.2.1

Combine the opposite terms in $x^{3} - 3 x^{2} + x^{2} i + x^{2} (- i) + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i - 2$ .

Tap for more steps...

Step 5.2.1.1

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

$x^{3} - 3 x^{2} + i x^{2} - i x^{2} + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i - 2$

Step 5.2.1.2

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

$x^{3} - 3 x^{2} + 0 + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i - 2$

Step 5.2.1.3

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

$x^{3} - 3 x^{2} + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 6 i + 2 i x - 6 i - 2$

Step 5.2.1.4

Subtract $6 i$ from $6 i$ .

$x^{3} - 3 x^{2} + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 2 i x + 0 - 2$

Step 5.2.1.5

Add $x^{3} - 3 x^{2} + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 2 i x$ and $0$ .

$x^{3} - 3 x^{2} + 3 x i + x - 5 x^{2} + 15 x - 5 x i + 6 x - 18 + 2 i x - 2$

Step 5.2.2

Subtract $5 x^{2}$ from $- 3 x^{2}$ .

$x^{3} - 8 x^{2} + 3 x i + x + 15 x - 5 x i + 6 x - 18 + 2 i x - 2$

Step 5.2.3

Subtract $5 x i$ from $3 x i$ .

$x^{3} - 8 x^{2} - 2 x i + x + 15 x + 6 x - 18 + 2 i x - 2$

Step 5.2.4

Combine the opposite terms in $x^{3} - 8 x^{2} - 2 x i + x + 15 x + 6 x - 18 + 2 i x - 2$ .

Tap for more steps...

Step 5.2.4.1

Reorder the factors in the terms $- 2 x i$ and $2 i x$ .

$x^{3} - 8 x^{2} - 2 i x + x + 15 x + 6 x - 18 + 2 i x - 2$

Step 5.2.4.2

Add $- 2 i x$ and $2 i x$ .

$x^{3} - 8 x^{2} + x + 15 x + 6 x - 18 + 0 - 2$

Step 5.2.4.3

Add $x^{3} - 8 x^{2} + x + 15 x + 6 x - 18$ and $0$ .

$x^{3} - 8 x^{2} + x + 15 x + 6 x - 18 - 2$

Step 5.2.5

Add $x$ and $15 x$ .

$x^{3} - 8 x^{2} + 16 x + 6 x - 18 - 2$

Step 5.2.6

Add $16 x$ and $6 x$ .

$x^{3} - 8 x^{2} + 22 x - 18 - 2$

Step 5.2.7

Subtract $2$ from $- 18$ .

$x^{3} - 8 x^{2} + 22 x - 20$