Trigonometry Examples

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

Step 1

Simplify $f (x)$ .

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Step 1.1

Expand $(x - 3) (x - 2)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $x$ by $x$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $- 3$ by $- 2$ .

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

Step 1.2.2

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

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

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Multiply $- 1$ by $2$ .

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

Step 1.3.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4

Multiply $i$ by $1$ .

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

Step 1.4

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

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

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

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

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Step 1.5.1.1.1.1

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

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

Step 1.5.1.1.1.2

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

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

Step 1.5.1.1.2

Add $2$ and $1$ .

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $x$ .

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

Step 1.5.1.3.2

Multiply $x$ by $x$ .

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

Step 1.5.1.4

Multiply $- 2$ by $- 5$ .

$f (x) = (x^{3} - 2 x^{2} + x^{2} i - 5 x^{2} + 10 x - 5 x i + 6 x + 6 \cdot - 2 + 6 i) (x - (2 + i))$

Step 1.5.1.5

Multiply $6$ by $- 2$ .

$f (x) = (x^{3} - 2 x^{2} + x^{2} i - 5 x^{2} + 10 x - 5 x i + 6 x - 12 + 6 i) (x - (2 + i))$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

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

$f (x) = (x^{3} - 7 x^{2} + x^{2} i + 10 x - 5 x i + 6 x - 12 + 6 i) (x - (2 + i))$

Step 1.5.2.2

Add $10 x$ and $6 x$ .

$f (x) = (x^{3} - 7 x^{2} + x^{2} i - 5 x i + 16 x - 12 + 6 i) (x - (2 + i))$

Step 1.5.3

Simplify each term.

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Step 1.5.3.1

Apply the distributive property.

$f (x) = (x^{3} - 7 x^{2} + x^{2} i - 5 x i + 16 x - 12 + 6 i) (x - 1 \cdot 2 - i)$

Step 1.5.3.2

Multiply $- 1$ by $2$ .

$f (x) = (x^{3} - 7 x^{2} + x^{2} i - 5 x i + 16 x - 12 + 6 i) (x - 2 - i)$

Step 1.6

Expand $(x^{3} - 7 x^{2} + x^{2} i - 5 x i + 16 x - 12 + 6 i) (x - 2 - i)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{3} x + x^{3} \cdot - 2 + x^{3} (- i) - 7 x^{2} x - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

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

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Step 1.7.1.1.1.1

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

Step 1.7.1.1.1.2

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

$f (x) = x^{3 + 1} + x^{3} \cdot - 2 + x^{3} (- i) - 7 x^{2} x - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.1.2

Add $3$ and $1$ .

$f (x) = x^{4} + x^{3} \cdot - 2 + x^{3} (- i) - 7 x^{2} x - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.2

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

$f (x) = x^{4} - 2 \cdot x^{3} + x^{3} (- i) - 7 x^{2} x - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 (x \cdot x^{2}) - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.3.2

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

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Step 1.7.1.3.2.1

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

Step 1.7.1.3.2.2

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{1 + 2} - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.3.3

Add $1$ and $2$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} - 7 x^{2} \cdot - 2 - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.4

Multiply $- 2$ by $- 7$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} - 7 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.5

Multiply $- 1$ by $- 7$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{2} i x + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.6

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

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Step 1.7.1.6.1

Move $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x \cdot x^{2} i + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.6.2

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

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Step 1.7.1.6.2.1

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

Step 1.7.1.6.2.2

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{1 + 2} i + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.6.3

Add $1$ and $2$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i + x^{2} i \cdot - 2 + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.7

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 \cdot (x^{2} i) + x^{2} i (- i) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.8

Multiply $x^{2} i (- i)$ .

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Step 1.7.1.8.1

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} (- (i i)) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.8.2

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

Step 1.7.1.8.3

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} (- i^{1 + 1}) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.8.4

Add $1$ and $1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} (- i^{2}) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.9

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} (1) - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.10

Multiply $- 1$ by $- 1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} \cdot 1 - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.11

Multiply $x^{2}$ by $1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x i x - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.12

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

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Step 1.7.1.12.1

Move $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 (x \cdot x) i - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.12.2

Multiply $x$ by $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i - 5 x i \cdot - 2 - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.13

Multiply $- 2$ by $- 5$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x i (- i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.14

Multiply $- 5 x i (- i)$ .

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Step 1.7.1.14.1

Multiply $- 1$ by $- 5$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i + 5 x i i + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.14.2

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i + 5 x (i i) + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.14.3

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

Step 1.7.1.14.4

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i + 5 x i^{1 + 1} + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.14.5

Add $1$ and $1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i + 5 x i^{2} + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.15

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i + 5 x \cdot - 1 + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.16

Multiply $- 1$ by $5$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x \cdot x + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.17

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

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Step 1.7.1.17.1

Move $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 (x \cdot x) + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.17.2

Multiply $x$ by $x$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} + 16 x \cdot - 2 + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.18

Multiply $- 2$ by $16$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x + 16 x (- i) - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.19

Multiply $- 1$ by $16$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x - 12 \cdot - 2 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.20

Multiply $- 12$ by $- 2$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 - 12 (- i) + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.21

Multiply $- 1$ by $- 12$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x + 6 i \cdot - 2 + 6 i (- i)$

Step 1.7.1.22

Multiply $- 2$ by $6$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6 i (- i)$

Step 1.7.1.23

Multiply $6 i (- i)$ .

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Step 1.7.1.23.1

Multiply $- 1$ by $6$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i - 6 i i$

Step 1.7.1.23.2

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i - 6 (i i)$

Step 1.7.1.23.3

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

Step 1.7.1.23.4

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i - 6 i^{1 + 1}$

Step 1.7.1.23.5

Add $1$ and $1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i - 6 i^{2}$

Step 1.7.1.24

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

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i - 6 \cdot - 1$

Step 1.7.1.25

Multiply $- 6$ by $- 1$ .

$f (x) = x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6$

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

Combine the opposite terms in $x^{4} - 2 x^{3} + x^{3} (- i) - 7 x^{3} + 14 x^{2} + 7 x^{2} i + x^{3} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6$ .

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Step 1.7.2.1.1

Reorder the factors in the terms $x^{3} (- i)$ and $x^{3} i$ .

$f (x) = x^{4} - 2 x^{3} - i x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i + i x^{3} - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6$

Step 1.7.2.1.2

Add $- i x^{3}$ and $i x^{3}$ .

$f (x) = x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i + 0 - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6$

Step 1.7.2.1.3

Add $x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i$ and $0$ .

$f (x) = x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 12 i + 6 i x - 12 i + 6$

Step 1.7.2.1.4

Subtract $12 i$ from $12 i$ .

$f (x) = x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 6 i x + 0 + 6$

Step 1.7.2.1.5

Add $x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 6 i x$ and $0$ .

$f (x) = x^{4} - 2 x^{3} - 7 x^{3} + 14 x^{2} + 7 x^{2} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.2

Subtract $7 x^{3}$ from $- 2 x^{3}$ .

$f (x) = x^{4} - 9 x^{3} + 14 x^{2} + 7 x^{2} i - 2 x^{2} i + x^{2} - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.3

Add $14 x^{2}$ and $x^{2}$ .

$f (x) = x^{4} - 9 x^{3} + 15 x^{2} + 7 x^{2} i - 2 x^{2} i - 5 x^{2} i + 10 x i - 5 x + 16 x^{2} - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.4

Add $15 x^{2}$ and $16 x^{2}$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} + 7 x^{2} i - 2 x^{2} i - 5 x^{2} i + 10 x i - 5 x - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.5

Subtract $2 x^{2} i$ from $7 x^{2} i$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} + 5 x^{2} i - 5 x^{2} i + 10 x i - 5 x - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.6

Combine the opposite terms in $x^{4} - 9 x^{3} + 31 x^{2} + 5 x^{2} i - 5 x^{2} i + 10 x i - 5 x - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$ .

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Step 1.7.2.6.1

Subtract $5 x^{2} i$ from $5 x^{2} i$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} + 0 + 10 x i - 5 x - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.6.2

Add $x^{4} - 9 x^{3} + 31 x^{2}$ and $0$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} + 10 x i - 5 x - 32 x - 16 x i - 12 x + 24 + 6 i x + 6$

Step 1.7.2.7

Subtract $16 x i$ from $10 x i$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 6 x i - 5 x - 32 x - 12 x + 24 + 6 i x + 6$

Step 1.7.2.8

Combine the opposite terms in $x^{4} - 9 x^{3} + 31 x^{2} - 6 x i - 5 x - 32 x - 12 x + 24 + 6 i x + 6$ .

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Step 1.7.2.8.1

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

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 6 i x - 5 x - 32 x - 12 x + 24 + 6 i x + 6$

Step 1.7.2.8.2

Add $- 6 i x$ and $6 i x$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 5 x - 32 x - 12 x + 24 + 0 + 6$

Step 1.7.2.8.3

Add $x^{4} - 9 x^{3} + 31 x^{2} - 5 x - 32 x - 12 x + 24$ and $0$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 5 x - 32 x - 12 x + 24 + 6$

Step 1.7.2.9

Subtract $32 x$ from $- 5 x$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 37 x - 12 x + 24 + 6$

Step 1.7.2.10

Subtract $12 x$ from $- 37 x$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 49 x + 24 + 6$

Step 1.7.2.11

Add $24$ and $6$ .

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 49 x + 30$

Step 2

The word linear is used for a straight line. A linear function is a function of a straight line, which means that the degree of a linear function must be $0$ or $1$ . In this case, The degree of $f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 49 x + 30$ is $4$ , which makes the function a nonlinear function.

$f (x) = x^{4} - 9 x^{3} + 31 x^{2} - 49 x + 30$ is not a linear function