Trigonometry Examples

$- 2$ , $0$ , $3 + 2 i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - 2$ was found by solving for $x$ when $x - (- 2) = y$ and $y = 0$ .

The factor is $x + 2$

Step 3

The root at $x = 0$ was found by solving for $x$ when $x - (0) = y$ and $y = 0$ .

The factor is $x$

Step 4

The root at $x = 3 + 2 i$ was found by solving for $x$ when $x - (3 + 2 i) = y$ and $y = 0$ .

The factor is $x - 3 - 2 i$

Step 5

The root at $x = 3 - 2 i$ was found by solving for $x$ when $x - (3 - 2 i) = y$ and $y = 0$ .

The factor is $x - 3 + 2 i$

Step 6

Combine all the factors into a single equation.

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

Step 7

Multiply all the factors to simplify the equation $y = (x + 2) (x) (x - 3 - 2 i) (x - 3 + 2 i)$ .

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Step 7.1

Simplify by multiplying through.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Multiply $x$ by $x$ .

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

Step 7.2

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

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

Step 7.3

Simplify terms.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

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

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Step 7.3.1.1.1

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

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Step 7.3.1.1.1.1

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

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

Step 7.3.1.1.1.2

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

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

Step 7.3.1.1.2

Add $2$ and $1$ .

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

Step 7.3.1.2

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

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

Step 7.3.1.3

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

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Step 7.3.1.3.1

Move $x$ .

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

Step 7.3.1.3.2

Multiply $x$ by $x$ .

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

Step 7.3.1.4

Multiply $- 3$ by $2$ .

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

Step 7.3.1.5

Multiply $- 2$ by $2$ .

$y = (x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} - 6 x - 4 x i) (x - 3 + 2 i)$

Step 7.3.2

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

$y = (x^{3} - x^{2} + x^{2} (- 2 i) - 6 x - 4 x i) (x - 3 + 2 i)$

Step 7.4

Expand $(x^{3} - x^{2} + x^{2} (- 2 i) - 6 x - 4 x i) (x - 3 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 7.5

Simplify terms.

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Step 7.5.1

Simplify each term.

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Step 7.5.1.1

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

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Step 7.5.1.1.1

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

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Step 7.5.1.1.1.1

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

Step 7.5.1.1.1.2

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

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

Step 7.5.1.1.2

Add $3$ and $1$ .

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

Step 7.5.1.2

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

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

Step 7.5.1.3

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

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

Step 7.5.1.4

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

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Step 7.5.1.4.1

Move $x$ .

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

Step 7.5.1.4.2

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

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Step 7.5.1.4.2.1

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

Step 7.5.1.4.2.2

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

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

Step 7.5.1.4.3

Add $1$ and $2$ .

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

Step 7.5.1.5

Multiply $- 3$ by $- 1$ .

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

Step 7.5.1.6

Multiply $2$ by $- 1$ .

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

Step 7.5.1.7

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

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Step 7.5.1.7.1

Move $x$ .

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

Step 7.5.1.7.2

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

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Step 7.5.1.7.2.1

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

Step 7.5.1.7.2.2

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

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

Step 7.5.1.7.3

Add $1$ and $2$ .

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

Step 7.5.1.8

Multiply $- 3$ by $- 2$ .

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

Step 7.5.1.9

Move $6$ to the left of $x^{2}$ .

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

Step 7.5.1.10

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

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Step 7.5.1.10.1

Multiply $2$ by $- 2$ .

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

Step 7.5.1.10.2

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

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

Step 7.5.1.10.3

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

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

Step 7.5.1.10.4

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

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

Step 7.5.1.10.5

Add $1$ and $1$ .

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

Step 7.5.1.11

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

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

Step 7.5.1.12

Multiply $- 4$ by $- 1$ .

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

Step 7.5.1.13

Move $4$ to the left of $x^{2}$ .

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

Step 7.5.1.14

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

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Step 7.5.1.14.1

Move $x$ .

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

Step 7.5.1.14.2

Multiply $x$ by $x$ .

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

Step 7.5.1.15

Multiply $- 3$ by $- 6$ .

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

Step 7.5.1.16

Multiply $2$ by $- 6$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x i x - 4 x i \cdot - 3 - 4 x i (2 i)$

Step 7.5.1.17

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

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Step 7.5.1.17.1

Move $x$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 (x \cdot x) i - 4 x i \cdot - 3 - 4 x i (2 i)$

Step 7.5.1.17.2

Multiply $x$ by $x$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i - 4 x i \cdot - 3 - 4 x i (2 i)$

Step 7.5.1.18

Multiply $- 3$ by $- 4$ .

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

Step 7.5.1.19

Multiply $- 4 x i (2 i)$ .

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Step 7.5.1.19.1

Multiply $2$ by $- 4$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x i i$

Step 7.5.1.19.2

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

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x (i i)$

Step 7.5.1.19.3

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

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x (i i)$

Step 7.5.1.19.4

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

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x i^{1 + 1}$

Step 7.5.1.19.5

Add $1$ and $1$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x i^{2}$

Step 7.5.1.20

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

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i - 8 x \cdot - 1$

Step 7.5.1.21

Multiply $- 1$ by $- 8$ .

$y = x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i + 8 x$

Step 7.5.2

Simplify by adding terms.

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Step 7.5.2.1

Combine the opposite terms in $x^{4} - 3 x^{3} + 2 x^{3} i - x^{3} + 3 x^{2} - 2 x^{2} i + x^{3} (- 2 i) + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i + 8 x$ .

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Step 7.5.2.1.1

Reorder the factors in the terms $2 x^{3} i$ and $x^{3} (- 2 i)$ .

$y = x^{4} - 3 x^{3} + 2 i x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i - 2 i x^{3} + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i + 8 x$

Step 7.5.2.1.2

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

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i + 0 + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i + 8 x$

Step 7.5.2.1.3

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

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 12 x i - 4 x^{2} i + 12 x i + 8 x$

Step 7.5.2.1.4

Add $- 12 x i$ and $12 x i$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 4 x^{2} i + 0 + 8 x$

Step 7.5.2.1.5

Add $x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 4 x^{2} i$ and $0$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 2 x^{2} i + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 4 x^{2} i + 8 x$

Step 7.5.2.2

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

$y = x^{4} - 4 x^{3} + 3 x^{2} - 2 x^{2} i + 6 x^{2} i + 4 x^{2} - 6 x^{2} + 18 x - 4 x^{2} i + 8 x$

Step 7.5.2.3

Add $3 x^{2}$ and $4 x^{2}$ .

$y = x^{4} - 4 x^{3} + 7 x^{2} - 2 x^{2} i + 6 x^{2} i - 6 x^{2} + 18 x - 4 x^{2} i + 8 x$

Step 7.5.2.4

Subtract $6 x^{2}$ from $7 x^{2}$ .

$y = x^{4} - 4 x^{3} + x^{2} - 2 x^{2} i + 6 x^{2} i + 18 x - 4 x^{2} i + 8 x$

Step 7.5.2.5

Add $- 2 x^{2} i$ and $6 x^{2} i$ .

$y = x^{4} - 4 x^{3} + x^{2} + 4 x^{2} i + 18 x - 4 x^{2} i + 8 x$

Step 7.5.2.6

Combine the opposite terms in $x^{4} - 4 x^{3} + x^{2} + 4 x^{2} i + 18 x - 4 x^{2} i + 8 x$ .

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Step 7.5.2.6.1

Subtract $4 x^{2} i$ from $4 x^{2} i$ .

$y = x^{4} - 4 x^{3} + x^{2} + 18 x + 0 + 8 x$

Step 7.5.2.6.2

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

$y = x^{4} - 4 x^{3} + x^{2} + 18 x + 8 x$

Step 7.5.2.7

Add $18 x$ and $8 x$ .

$y = x^{4} - 4 x^{3} + x^{2} + 26 x$

Step 8