Precalculus Examples

$1 z = 2 (\cos (34) + i \sin (34))$ , $2 z = 5 (\cos (10) + i \sin (10))$

Step 1

Simplify the left side.

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Step 1.1

Multiply $z$ by $1$ .

$z = 2 (\cos (34) + i \sin (34))$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 2 (\cos (34) + i \sin (34))$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2

Simplify the right side.

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Step 2.1

Simplify $2 (\cos (34) + i \sin (34))$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Evaluate $\cos (34)$ .

$z = 2 (0.82903757 + i \sin (34))$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2.1.1.2

Evaluate $\sin (34)$ .

$z = 2 (0.82903757 + i \cdot 0.5591929)$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2.1.1.3

Move $0.5591929$ to the left of $i$ .

$z = 2 (0.82903757 + 0.5591929 i)$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 2 (0.82903757 + 0.5591929 i)$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2.1.2

Simplify by multiplying through.

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Step 2.1.2.1

Apply the distributive property.

$z = 2 \cdot 0.82903757 + 2 (0.5591929 i)$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2.1.2.2

Multiply.

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Step 2.1.2.2.1

Multiply $2$ by $0.82903757$ .

$z = 1.65807514 + 2 (0.5591929 i)$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 2.1.2.2.2

Multiply $0.5591929$ by $2$ .

$z = 1.65807514 + 1.1183858 i$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 1.65807514 + 1.1183858 i$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 1.65807514 + 1.1183858 i$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 1.65807514 + 1.1183858 i$

$2 z = 5 (\cos (10) + i \sin (10))$

$z = 1.65807514 + 1.1183858 i$

$2 z = 5 (\cos (10) + i \sin (10))$

Step 3

Simplify.

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Step 3.1

Evaluate $\sin (10)$ .

$2 z = 5 (\cos (10) + i \cdot 0.17364817)$

$z = 1.65807514 + 1.1183858 i$

Step 3.2

Move $0.17364817$ to the left of $i$ .

$2 z = 5 (\cos (10) + 0.17364817 \cdot i)$

$z = 1.65807514 + 1.1183858 i$

Step 3.3

Evaluate $\cos (10)$ .

$2 z = 5 (0.98480775 + 0.17364817 i)$

$z = 1.65807514 + 1.1183858 i$

Step 3.4

Apply the distributive property.

$2 z = 5 \cdot 0.98480775 + 5 (0.17364817 i)$

$z = 1.65807514 + 1.1183858 i$

Step 3.5

Multiply $5$ by $0.98480775$ .

$2 z = 4.92403876 + 5 (0.17364817 i)$

$z = 1.65807514 + 1.1183858 i$

Step 3.6

Multiply $0.17364817$ by $5$ .

$2 z = 4.92403876 + 0.86824088 i$

$z = 1.65807514 + 1.1183858 i$

$2 z = 4.92403876 + 0.86824088 i$

$z = 1.65807514 + 1.1183858 i$

Step 4

Divide each term in $2 z = 4.92403876 + 0.86824088 i$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 z = 4.92403876 + 0.86824088 i$ by $2$ .

$\frac{2 z}{2} = \frac{4.92403876}{2} + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{4.92403876}{2} + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

Step 4.2.1.2

Divide $z$ by $1$ .

$z = \frac{4.92403876}{2} + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

$z = \frac{4.92403876}{2} + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

$z = \frac{4.92403876}{2} + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Divide $4.92403876$ by $2$ .

$z = 2.46201938 + \frac{0.86824088 i}{2}$

$z = 1.65807514 + 1.1183858 i$

Step 4.3.1.2

Factor $0.86824088$ out of $0.86824088 i$ .

$z = 2.46201938 + \frac{0.86824088 (i)}{2}$

$z = 1.65807514 + 1.1183858 i$

Step 4.3.1.3

Factor $2$ out of $2$ .

$z = 2.46201938 + \frac{0.86824088 (i)}{2 (1)}$

$z = 1.65807514 + 1.1183858 i$

Step 4.3.1.4

Separate fractions.

$z = 2.46201938 + \frac{0.86824088}{2} \cdot \frac{i}{1}$

$z = 1.65807514 + 1.1183858 i$

Step 4.3.1.5

Divide $0.86824088$ by $2$ .

$z = 2.46201938 + 0.43412044 (\frac{i}{1})$

$z = 1.65807514 + 1.1183858 i$

Step 4.3.1.6

Divide $i$ by $1$ .

$z = 2.46201938 + 0.43412044 i$

$z = 1.65807514 + 1.1183858 i$

$z = 2.46201938 + 0.43412044 i$

$z = 1.65807514 + 1.1183858 i$

$z = 2.46201938 + 0.43412044 i$

$z = 1.65807514 + 1.1183858 i$

$z = 2.46201938 + 0.43412044 i$

$z = 1.65807514 + 1.1183858 i$

Step 5

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(z - (1.65807514 + 1.1183858 i)) (z - (1.65807514 - 1.1183858 i)) (z - (2.46201938 + 0.43412044 i)) (z - (2.46201938 - 0.43412044 i)) = 0$

Step 6

Simplify.

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Step 6.1

Expand $(z - 1.65807514 - 1.1183858 i) (z - 1.65807514 + 1.1183858 i)$ by multiplying each term in the first expression by each term in the second expression.

$(z \cdot z + z \cdot - 1.65807514 + z (1.1183858 i) - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i z - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine the opposite terms in $z \cdot z + z \cdot - 1.65807514 + z (1.1183858 i) - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i z - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)$ .

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Step 6.2.1.1

Reorder the factors in the terms $z (1.1183858 i)$ and $- 1.1183858 i z$ .

$(z \cdot z + z \cdot - 1.65807514 + 1.1183858 i z - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i z - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.1.2

Subtract $1.1183858 i z$ from $1.1183858 i z$ .

$(z \cdot z + z \cdot - 1.65807514 + 0 i z - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Multiply $z$ by $z$ .

$(z^{2} + z \cdot - 1.65807514 + 0 i z - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.2

Move $- 1.65807514$ to the left of $z$ .

$(z^{2} - 1.65807514 \cdot z + 0 i z - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.3

Multiply $0$ by $i$ .

$(z^{2} - 1.65807514 z + 0 z - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.4

Multiply $0$ by $z$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z - 1.65807514 \cdot - 1.65807514 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.5

Multiply $- 1.65807514$ by $- 1.65807514$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.65807514 (1.1183858 i) - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.6

Multiply $1.1183858$ by $- 1.65807514$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i - 1.1183858 i \cdot - 1.65807514 - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.7

Multiply $- 1.65807514$ by $- 1.1183858$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.1183858 i (1.1183858 i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.8

Multiply $- 1.1183858 i (1.1183858 i)$ .

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Step 6.2.2.8.1

Multiply $1.1183858$ by $- 1.1183858$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 i i) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.8.2

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

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 (i i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.8.3

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

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 (i i)) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.8.4

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

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 i^{1 + 1}) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.8.5

Add $1$ and $1$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 i^{2}) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.9

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

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i - 1.25078681 \cdot - 1) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.2.10

Multiply $- 1.25078681$ by $- 1$ .

$(z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3

Simplify by adding terms.

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Step 6.2.3.1

Combine the opposite terms in $z^{2} - 1.65807514 z + 0 - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i + 1.25078681$ .

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Step 6.2.3.1.1

Add $z^{2} - 1.65807514 z$ and $0$ .

$(z^{2} - 1.65807514 z - 1.65807514 z + 2.74921318 - 1.8543677 i + 1.8543677 i + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3.1.2

Add $- 1.8543677 i$ and $1.8543677 i$ .

$(z^{2} - 1.65807514 z - 1.65807514 z + 2.74921318 + 0 i + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3.2

Simplify the expression.

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Step 6.2.3.2.1

Multiply $0$ by $i$ .

$(z^{2} - 1.65807514 z - 1.65807514 z + 2.74921318 + 0 + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3.2.2

Add $z^{2} - 1.65807514 z - 1.65807514 z + 2.74921318$ and $0$ .

$(z^{2} - 1.65807514 z - 1.65807514 z + 2.74921318 + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3.3

Subtract $1.65807514 z$ from $- 1.65807514 z$ .

$(z^{2} - 3.31615029 z + 2.74921318 + 1.25078681) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.2.3.4

Add $2.74921318$ and $1.25078681$ .

$(z^{2} - 3.31615029 z + 4) (z - 2.46201938 - 0.43412044 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.3

Expand $(z^{2} - 3.31615029 z + 4) (z - 2.46201938 - 0.43412044 i)$ by multiplying each term in the first expression by each term in the second expression.

$(z^{2} z + z^{2} \cdot - 2.46201938 + z^{2} (- 0.43412044 i) - 3.31615029 z \cdot z - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

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Step 6.4.1.1.1.1

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

Step 6.4.1.1.1.2

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

$(z^{2 + 1} + z^{2} \cdot - 2.46201938 + z^{2} (- 0.43412044 i) - 3.31615029 z \cdot z - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.1.2

Add $2$ and $1$ .

$(z^{3} + z^{2} \cdot - 2.46201938 + z^{2} (- 0.43412044 i) - 3.31615029 z \cdot z - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.2

Move $- 2.46201938$ to the left of $z^{2}$ .

$(z^{3} - 2.46201938 \cdot z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z \cdot z - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.3

Multiply $z$ by $z$ by adding the exponents.

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Step 6.4.1.3.1

Move $z$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 (z \cdot z) - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.3.2

Multiply $z$ by $z$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z^{2} - 3.31615029 z \cdot - 2.46201938 - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.4

Multiply $- 2.46201938$ by $- 3.31615029$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z^{2} + 8.16442628 z - 3.31615029 z (- 0.43412044 i) + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.5

Multiply $- 0.43412044$ by $- 3.31615029$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z^{2} + 8.16442628 z + 1.43960863 z i + 4 z + 4 \cdot - 2.46201938 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.6

Multiply $4$ by $- 2.46201938$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z^{2} + 8.16442628 z + 1.43960863 z i + 4 z - 9.84807753 + 4 (- 0.43412044 i)) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.1.7

Multiply $- 0.43412044$ by $4$ .

$(z^{3} - 2.46201938 z^{2} + z^{2} (- 0.43412044 i) - 3.31615029 z^{2} + 8.16442628 z + 1.43960863 z i + 4 z - 9.84807753 - 1.73648177 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Subtract $3.31615029 z^{2}$ from $- 2.46201938 z^{2}$ .

$(z^{3} - 5.77816967 z^{2} + z^{2} (- 0.43412044 i) + 8.16442628 z + 1.43960863 z i + 4 z - 9.84807753 - 1.73648177 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.4.2.2

Add $8.16442628 z$ and $4 z$ .

$(z^{3} - 5.77816967 z^{2} + z^{2} (- 0.43412044 i) + 1.43960863 z i + 12.16442628 z - 9.84807753 - 1.73648177 i) (z - 2.46201938 + 0.43412044 i) = 0$

Step 6.5

Expand $(z^{3} - 5.77816967 z^{2} + z^{2} (- 0.43412044 i) + 1.43960863 z i + 12.16442628 z - 9.84807753 - 1.73648177 i) (z - 2.46201938 + 0.43412044 i)$ by multiplying each term in the first expression by each term in the second expression.

$z^{3} z + z^{3} \cdot - 2.46201938 + z^{3} (0.43412044 i) - 5.77816967 z^{2} z - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6

Simplify terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

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

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Step 6.6.1.1.1.1

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

Step 6.6.1.1.1.2

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

$z^{3 + 1} + z^{3} \cdot - 2.46201938 + z^{3} (0.43412044 i) - 5.77816967 z^{2} z - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.1.2

Add $3$ and $1$ .

$z^{4} + z^{3} \cdot - 2.46201938 + z^{3} (0.43412044 i) - 5.77816967 z^{2} z - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.2

Move $- 2.46201938$ to the left of $z^{3}$ .

$z^{4} - 2.46201938 \cdot z^{3} + z^{3} (0.43412044 i) - 5.77816967 z^{2} z - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.3

Move $0.43412044$ to the left of $z^{3}$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 \cdot (z^{3} i) - 5.77816967 z^{2} z - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.4

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

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Step 6.6.1.4.1

Move $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 (z \cdot z^{2}) - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.4.2

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

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Step 6.6.1.4.2.1

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

Step 6.6.1.4.2.2

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{1 + 2} - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.4.3

Add $1$ and $2$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} - 5.77816967 z^{2} \cdot - 2.46201938 - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.5

Multiply $- 2.46201938$ by $- 5.77816967$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 5.77816967 z^{2} (0.43412044 i) + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.6

Multiply $0.43412044$ by $- 5.77816967$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{2} (- 0.43412044 i) z + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.7

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

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Step 6.6.1.7.1

Move $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z \cdot z^{2} (- 0.43412044 i) + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.7.2

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

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Step 6.6.1.7.2.1

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

Step 6.6.1.7.2.2

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{1 + 2} (- 0.43412044 i) + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.7.3

Add $1$ and $2$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + z^{2} (- 0.43412044 i) \cdot - 2.46201938 + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.8

Multiply $- 2.46201938$ by $- 0.43412044$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + z^{2} (1.06881294 i) + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.9

Move $1.06881294$ to the left of $z^{2}$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 \cdot (z^{2} i) + z^{2} (- 0.43412044 i) (0.43412044 i) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.10

Multiply $z^{2} (- 0.43412044 i) (0.43412044 i)$ .

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Step 6.6.1.10.1

Multiply $0.43412044$ by $- 0.43412044$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} (- 0.18846056 i) i + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.10.2

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} (- 0.18846056 (i i)) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.10.3

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

Step 6.6.1.10.4

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} (- 0.18846056 i^{1 + 1}) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.10.5

Add $1$ and $1$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} (- 0.18846056 i^{2}) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.11

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} (- 0.18846056 \cdot - 1) + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.12

Multiply $- 0.18846056$ by $- 1$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + z^{2} \cdot 0.18846056 + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.13

Move $0.18846056$ to the left of $z^{2}$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 \cdot z^{2} + 1.43960863 z i z + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.14

Multiply $z$ by $z$ by adding the exponents.

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Step 6.6.1.14.1

Move $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 (z \cdot z) i + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.14.2

Multiply $z$ by $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i + 1.43960863 z i \cdot - 2.46201938 + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.15

Multiply $- 2.46201938$ by $1.43960863$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 1.43960863 z i (0.43412044 i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.16

Multiply $1.43960863 z i (0.43412044 i)$ .

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Step 6.6.1.16.1

Multiply $0.43412044$ by $1.43960863$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z i i + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.16.2

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z (i i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.16.3

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z (i i) + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.16.4

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z i^{1 + 1} + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.16.5

Add $1$ and $1$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z i^{2} + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.17

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i + 0.62496354 z \cdot - 1 + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.18

Multiply $- 1$ by $0.62496354$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z \cdot z + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.19

Multiply $z$ by $z$ by adding the exponents.

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Step 6.6.1.19.1

Move $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 (z \cdot z) + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.19.2

Multiply $z$ by $z$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} + 12.16442628 z \cdot - 2.46201938 + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.20

Multiply $- 2.46201938$ by $12.16442628$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 12.16442628 z (0.43412044 i) - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.21

Multiply $0.43412044$ by $12.16442628$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z - 9.84807753 \cdot - 2.46201938 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.22

Multiply $- 9.84807753$ by $- 2.46201938$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 9.84807753 (0.43412044 i) - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.23

Multiply $0.43412044$ by $- 9.84807753$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z - 1.73648177 i \cdot - 2.46201938 - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.24

Multiply $- 2.46201938$ by $- 1.73648177$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 1.73648177 i (0.43412044 i) = 0$

Step 6.6.1.25

Multiply $- 1.73648177 i (0.43412044 i)$ .

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Step 6.6.1.25.1

Multiply $0.43412044$ by $- 1.73648177$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 i i = 0$

Step 6.6.1.25.2

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 (i i) = 0$

Step 6.6.1.25.3

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 (i i) = 0$

Step 6.6.1.25.4

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 i^{1 + 1} = 0$

Step 6.6.1.25.5

Add $1$ and $1$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 i^{2} = 0$

Step 6.6.1.26

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

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i - 0.75384224 \cdot - 1 = 0$

Step 6.6.1.27

Multiply $- 0.75384224$ by $- 1$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.2

Combine the opposite terms in $z^{4} - 2.46201938 z^{3} + 0.43412044 z^{3} i - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + z^{3} (- 0.43412044 i) + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224$ .

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Step 6.6.2.1

Reorder the factors in the terms $0.43412044 z^{3} i$ and $z^{3} (- 0.43412044 i)$ .

$z^{4} - 2.46201938 z^{3} + 0.43412044 i z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i - 0.43412044 i z^{3} + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.2.2

Subtract $0.43412044 i z^{3}$ from $0.43412044 i z^{3}$ .

$z^{4} - 2.46201938 z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + 0 i z^{3} + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.3

Simplify each term.

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Step 6.6.3.1

Multiply $0$ by $i$ .

$z^{4} - 2.46201938 z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + 0 z^{3} + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.3.2

Multiply $0$ by $z^{3}$ .

$z^{4} - 2.46201938 z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + 0 + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4

Simplify by adding terms.

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Step 6.6.4.1

Add $z^{4} - 2.46201938 z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i$ and $0$ .

$z^{4} - 2.46201938 z^{3} - 5.77816967 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4.2

Subtract $5.77816967 z^{3}$ from $- 2.46201938 z^{3}$ .

$z^{4} - 8.24018905 z^{3} + 14.22596572 z^{2} - 2.50842158 z^{2} i + 1.06881294 z^{2} i + 0.18846056 z^{2} + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4.3

Add $14.22596572 z^{2}$ and $0.18846056 z^{2}$ .

$z^{4} - 8.24018905 z^{3} + 14.41442628 z^{2} - 2.50842158 z^{2} i + 1.06881294 z^{2} i + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z + 12.16442628 z^{2} - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4.4

Add $14.41442628 z^{2}$ and $12.16442628 z^{2}$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} - 2.50842158 z^{2} i + 1.06881294 z^{2} i + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4.5

Add $- 2.50842158 z^{2} i$ and $1.06881294 z^{2} i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} - 1.43960863 z^{2} i + 1.43960863 z^{2} i - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.4.6

Add $- 1.43960863 z^{2} i$ and $1.43960863 z^{2} i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} + 0 z^{2} i - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.5

Simplify each term.

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Step 6.6.5.1

Multiply $0$ by $z^{2}$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} + 0 i - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.5.2

Multiply $0$ by $i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} + 0 - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.6

Simplify by adding terms.

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Step 6.6.6.1

Add $z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2}$ and $0$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} - 3.54434436 z i - 0.62496354 z - 29.9490533 z + 5.28082614 z i - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.6.6.2

Add $- 3.54434436 z i$ and $5.28082614 z i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} + 1.73648177 z i - 0.62496354 z - 29.9490533 z - 9.84807753 z + 24.24615775 - 4.27525179 i - 1.73648177 i z + 4.27525179 i + 0.75384224 = 0$

Step 6.7

Subtract $1.73648177 i z$ from $1.73648177 z i$ .

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Step 6.7.1

Move $i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} + 1.73648177 z i - 1.73648177 z i - 0.62496354 z - 29.9490533 z - 9.84807753 z + 24.24615775 - 4.27525179 i + 4.27525179 i + 0.75384224 = 0$

Step 6.7.2

Subtract $1.73648177 z i$ from $1.73648177 z i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} 0 z i - 0.62496354 z - 29.9490533 z - 9.84807753 z + 24.24615775 - 4.27525179 i + 4.27525179 i + 0.75384224 = 0$

Step 6.8

Subtract $29.9490533 z$ from $- 0.62496354 z$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} 0 z i - 30.57401684 z - 9.84807753 z + 24.24615775 - 4.27525179 i + 4.27525179 i + 0.75384224 = 0$

Step 6.9

Subtract $9.84807753 z$ from $- 30.57401684 z$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} 0 z i - 40.42209437 z + 24.24615775 - 4.27525179 i + 4.27525179 i + 0.75384224 = 0$

Step 6.10

Add $24.24615775$ and $0.75384224$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} 0 z i - 40.42209437 z + 25 - 4.27525179 i + 4.27525179 i = 0$

Step 6.11

Add $- 4.27525179 i$ and $4.27525179 i$ .

$z^{4} - 8.24018905 z^{3} + 26.57885257 z^{2} 0 z i - 40.42209437 z + 250 i = 0$