Precalculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x (- 2 i) + 3 x + 3 \cdot - 3 + 3 (- 2 i)$ .

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Step 2.1.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 2.1.2

Add $- 3 x$ and $3 x$ .

$(x \cdot x + x (- 2 i) + 0 + 3 \cdot - 3 + 3 (- 2 i)) (x - 3 + 2 i)$

Step 2.1.3

Add $x \cdot x + x (- 2 i)$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

Multiply $3$ by $- 3$ .

$(x^{2} + x (- 2 i) - 9 + 3 (- 2 i)) (x - 3 + 2 i)$

Step 2.2.3

Multiply $- 2$ by $3$ .

$(x^{2} + x (- 2 i) - 9 - 6 i) (x - 3 + 2 i)$

Step 3

Expand $(x^{2} + x (- 2 i) - 9 - 6 i) (x - 3 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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Step 4.1.1.1.1

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

$x^{2} x^{1} + x^{2} \cdot - 3 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.1.1.2

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

$x^{2 + 1} + x^{2} \cdot - 3 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.1.2

Add $2$ and $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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Step 4.1.4.1

Move $x$ .

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

Step 4.1.4.2

Multiply $x$ by $x$ .

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

Step 4.1.5

Multiply $- 3$ by $- 2$ .

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

Step 4.1.6

Move $6$ to the left of $x$ .

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

Step 4.1.7

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

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Step 4.1.7.1

Multiply $2$ by $- 2$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i) i - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.7.2

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 (i^{1} i)) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.7.3

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 (i^{1} i^{1})) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.7.4

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i^{1 + 1}) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.7.5

Add $1$ and $1$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i^{2}) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.8

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 \cdot - 1) - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.9

Multiply $- 4$ by $- 1$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x \cdot 4 - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.10

Move $4$ to the left of $x$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 \cdot x - 9 x - 9 \cdot - 3 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.11

Multiply $- 9$ by $- 3$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 9 (2 i) - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.12

Multiply $2$ by $- 9$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x - 6 i \cdot - 3 - 6 i (2 i)$

Step 4.1.13

Multiply $- 3$ by $- 6$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 6 i (2 i)$

Step 4.1.14

Multiply $- 6 i (2 i)$ .

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Step 4.1.14.1

Multiply $2$ by $- 6$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 i i$

Step 4.1.14.2

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 (i^{1} i)$

Step 4.1.14.3

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 (i^{1} i^{1})$

Step 4.1.14.4

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 i^{1 + 1}$

Step 4.1.14.5

Add $1$ and $1$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 i^{2}$

Step 4.1.15

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

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i - 12 \cdot - 1$

Step 4.1.16

Multiply $- 12$ by $- 1$ .

$x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Combine the opposite terms in $x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$ .

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Step 4.2.1.1

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

$x^{3} - 3 x^{2} + 2 i x^{2} - 2 i x^{2} + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$

Step 4.2.1.2

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

$x^{3} - 3 x^{2} + 0 + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$

Step 4.2.1.3

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

$x^{3} - 3 x^{2} + 6 x i + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$

Step 4.2.1.4

Reorder the factors in the terms $6 x i$ and $- 6 i x$ .

$x^{3} - 3 x^{2} + 6 i x + 4 x - 9 x + 27 - 18 i - 6 i x + 18 i + 12$

Step 4.2.1.5

Subtract $6 i x$ from $6 i x$ .

$x^{3} - 3 x^{2} + 4 x - 9 x + 27 - 18 i + 0 + 18 i + 12$

Step 4.2.1.6

Add $x^{3} - 3 x^{2} + 4 x - 9 x + 27 - 18 i$ and $0$ .

$x^{3} - 3 x^{2} + 4 x - 9 x + 27 - 18 i + 18 i + 12$

Step 4.2.1.7

Add $- 18 i$ and $18 i$ .

$x^{3} - 3 x^{2} + 4 x - 9 x + 27 + 0 + 12$

Step 4.2.1.8

Add $x^{3} - 3 x^{2} + 4 x - 9 x + 27$ and $0$ .

$x^{3} - 3 x^{2} + 4 x - 9 x + 27 + 12$

Step 4.2.2

Subtract $9 x$ from $4 x$ .

$x^{3} - 3 x^{2} - 5 x + 27 + 12$

Step 4.2.3

Add $27$ and $12$ .

$x^{3} - 3 x^{2} - 5 x + 39$