Precalculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

Multiply $2$ by $- 2$ .

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

Step 2.2.3

Multiply $- 1$ by $2$ .

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

Step 3

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

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

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 $1$ .

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

Step 4.1.1.1.2

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

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

Step 4.1.1.2

Add $2$ and $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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Step 4.1.3.1

Move $x$ .

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

Step 4.1.3.2

Multiply $x$ by $x$ .

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

Step 4.1.4

Multiply $- 2$ by $- 1$ .

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

Step 4.1.5

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

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

Step 4.1.6

Multiply $x (- i) i$ .

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

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

Step 4.1.6.4

Add $1$ and $1$ .

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

Step 4.1.7

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

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

Step 4.1.8

Multiply $- 1$ by $- 1$ .

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

Step 4.1.9

Multiply $x$ by $1$ .

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

Step 4.1.10

Multiply $- 4$ by $- 2$ .

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

Step 4.1.11

Multiply $- 2$ by $- 2$ .

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

Step 4.1.12

Multiply $- 2 i i$ .

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Step 4.1.12.1

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

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

Step 4.1.12.2

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

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

Step 4.1.12.3

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

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

Step 4.1.12.4

Add $1$ and $1$ .

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

Step 4.1.13

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

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

Step 4.1.14

Multiply $- 2$ by $- 1$ .

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

Step 4.2

Simplify by adding terms.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

Subtract $2 i x$ from $2 i x$ .

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

Step 4.2.1.6

Add $x^{3} - 2 x^{2} + x - 4 x + 8 - 4 i$ and $0$ .

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

Step 4.2.1.7

Add $- 4 i$ and $4 i$ .

$(x^{3} - 2 x^{2} + x - 4 x + 8 + 0 + 2) (x - 3)$

Step 4.2.1.8

Add $x^{3} - 2 x^{2} + x - 4 x + 8$ and $0$ .

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

Step 4.2.2

Subtract $4 x$ from $x$ .

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

Step 4.2.3

Add $8$ and $2$ .

$(x^{3} - 2 x^{2} - 3 x + 10) (x - 3)$

Step 5

Expand $(x^{3} - 2 x^{2} - 3 x + 10) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot - 3 - 2 x^{2} x - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 6.1.1.1

Multiply $x^{3}$ by $x$ .

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Step 6.1.1.1.1

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

$x^{3} x^{1} + x^{3} \cdot - 3 - 2 x^{2} x - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.1.1.2

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

$x^{3 + 1} + x^{3} \cdot - 3 - 2 x^{2} x - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.1.2

Add $3$ and $1$ .

$x^{4} + x^{3} \cdot - 3 - 2 x^{2} x - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.2

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

$x^{4} - 3 \cdot x^{3} - 2 x^{2} x - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.3

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

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Step 6.1.3.1

Move $x$ .

$x^{4} - 3 x^{3} - 2 (x \cdot x^{2}) - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.3.2

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

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Step 6.1.3.2.1

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

$x^{4} - 3 x^{3} - 2 (x^{1} x^{2}) - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.3.2.2

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

$x^{4} - 3 x^{3} - 2 x^{1 + 2} - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.3.3

Add $1$ and $2$ .

$x^{4} - 3 x^{3} - 2 x^{3} - 2 x^{2} \cdot - 3 - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.4

Multiply $- 3$ by $- 2$ .

$x^{4} - 3 x^{3} - 2 x^{3} + 6 x^{2} - 3 x \cdot x - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.5

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

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Step 6.1.5.1

Move $x$ .

$x^{4} - 3 x^{3} - 2 x^{3} + 6 x^{2} - 3 (x \cdot x) - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.5.2

Multiply $x$ by $x$ .

$x^{4} - 3 x^{3} - 2 x^{3} + 6 x^{2} - 3 x^{2} - 3 x \cdot - 3 + 10 x + 10 \cdot - 3$

Step 6.1.6

Multiply $- 3$ by $- 3$ .

$x^{4} - 3 x^{3} - 2 x^{3} + 6 x^{2} - 3 x^{2} + 9 x + 10 x + 10 \cdot - 3$

Step 6.1.7

Multiply $10$ by $- 3$ .

$x^{4} - 3 x^{3} - 2 x^{3} + 6 x^{2} - 3 x^{2} + 9 x + 10 x - 30$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

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

$x^{4} - 5 x^{3} + 6 x^{2} - 3 x^{2} + 9 x + 10 x - 30$

Step 6.2.2

Subtract $3 x^{2}$ from $6 x^{2}$ .

$x^{4} - 5 x^{3} + 3 x^{2} + 9 x + 10 x - 30$

Step 6.2.3

Add $9 x$ and $10 x$ .

$x^{4} - 5 x^{3} + 3 x^{2} + 19 x - 30$