Precalculus Examples

$(x - (1 + i)) (x - (1 - i))$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Multiply $- 1$ by $1$ .

$(x - 1 - i) (x - (1 - i))$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $- 1$ by $1$ .

$(x - 1 - i) (x - 1 - - i)$

Step 1.2.3

Multiply $- 1$ by $- 1$ .

$(x - 1 - i) (x - 1 + 1 i)$

Step 1.2.4

Multiply $i$ by $1$ .

$(x - 1 - i) (x - 1 + i)$

Step 2

Expand $(x - 1 - i) (x - 1 + i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 1 + x i - 1 x - 1 \cdot - 1 - 1 i - i x - i \cdot - 1 - i i$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x i - 1 x - 1 \cdot - 1 - 1 i - i x - i \cdot - 1 - i i$ .

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Step 3.1.1

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

$x \cdot x + x \cdot - 1 + i x - 1 x - 1 \cdot - 1 - 1 i - i x - i \cdot - 1 - i i$

Step 3.1.2

Subtract $i x$ from $i x$ .

$x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.1.3

Add $x \cdot x + x \cdot - 1$ and $0$ .

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.2.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.2.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.2.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - 1 i - i \cdot - 1 - i i$

Step 3.2.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - 1 i - i \cdot - 1 - i i$

Step 3.2.6

Rewrite $- 1 i$ as $- i$ .

$x^{2} - x - x + 1 - i - i \cdot - 1 - i i$

Step 3.2.7

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - i + 1 i - i i$

Step 3.2.8

Multiply $i$ by $1$ .

$x^{2} - x - x + 1 - i + i - i i$

Step 3.2.9

Multiply $- i i$ .

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Step 3.2.9.1

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

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

Step 3.2.9.2

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

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

Step 3.2.9.3

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

$x^{2} - x - x + 1 - i + i - i^{1 + 1}$

Step 3.2.9.4

Add $1$ and $1$ .

$x^{2} - x - x + 1 - i + i - i^{2}$

Step 3.2.10

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

$x^{2} - x - x + 1 - i + i - - 1$

Step 3.2.11

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - i + i + 1$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - x - x + 1 - i + i + 1$ .

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Step 3.3.1.1

Add $- i$ and $i$ .

$x^{2} - x - x + 1 + 0 + 1$

Step 3.3.1.2

Add $x^{2} - x - x + 1$ and $0$ .

$x^{2} - x - x + 1 + 1$

Step 3.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 + 1$

Step 3.3.3

Add $1$ and $1$ .

$x^{2} - 2 x + 2$