Precalculus Examples

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

Step 1

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

$x \cdot x + x \cdot - 4 + x i - 4 x - 4 \cdot - 4 - 4 i - i x - i \cdot - 4 - i i$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x \cdot - 4 + x i - 4 x - 4 \cdot - 4 - 4 i - i x - i \cdot - 4 - i i$ .

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Step 2.1.1

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

$x \cdot x + x \cdot - 4 + i x - 4 x - 4 \cdot - 4 - 4 i - i x - i \cdot - 4 - i i$

Step 2.1.2

Subtract $i x$ from $i x$ .

$x \cdot x + x \cdot - 4 + 0 - 4 x - 4 \cdot - 4 - 4 i - i \cdot - 4 - i i$

Step 2.1.3

Add $x \cdot x + x \cdot - 4$ and $0$ .

$x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 - 4 i - i \cdot - 4 - i i$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

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

$x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 - 4 i - i \cdot - 4 - i i$

Step 2.2.3

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x - 4 x + 16 - 4 i - i \cdot - 4 - i i$

Step 2.2.4

Multiply $- 4$ by $- 1$ .

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

Step 2.2.5

Multiply $- i i$ .

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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

Step 2.2.5.3

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

$x^{2} - 4 x - 4 x + 16 - 4 i + 4 i - i^{1 + 1}$

Step 2.2.5.4

Add $1$ and $1$ .

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

Step 2.2.6

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

$x^{2} - 4 x - 4 x + 16 - 4 i + 4 i - - 1$

Step 2.2.7

Multiply $- 1$ by $- 1$ .

$x^{2} - 4 x - 4 x + 16 - 4 i + 4 i + 1$

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Combine the opposite terms in $x^{2} - 4 x - 4 x + 16 - 4 i + 4 i + 1$ .

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Step 2.3.1.1

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

$x^{2} - 4 x - 4 x + 16 + 0 + 1$

Step 2.3.1.2

Add $x^{2} - 4 x - 4 x + 16$ and $0$ .

$x^{2} - 4 x - 4 x + 16 + 1$

Step 2.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 8 x + 16 + 1$

Step 2.3.3

Add $16$ and $1$ .

$x^{2} - 8 x + 17$