Algebra Examples

- 8 + 5 i

$- 8 + 5 i$ ,

$7 i$

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

$x = - 8 + 5 i$ was found by solving for

$x$ when

$x - (- 8 + 5 i) = y$ and

$y = 0$ .

The factor is

$x + 8 - 5 i$

Step 3

The root at

$x = - 8 - 5 i$ was found by solving for

$x$ when

$x - (- 8 - 5 i) = y$ and

$y = 0$ .

The factor is

$x + 8 + 5 i$

Step 4

The root at

$x = 7 i$ was found by solving for

$x$ when

$x - (7 i) = y$ and

$y = 0$ .

The factor is

$x - 7 i$

Step 5

The root at

$x = - 7 i$ was found by solving for

$x$ when

$x - (- 7 i) = y$ and

$y = 0$ .

The factor is

$x + 7 i$

Step 6

Combine all the factors into a single equation.

$y = (x + 8 - 5 i) (x + 8 + 5 i) (x - 7 i) (x + 7 i)$

Step 7

Multiply all the factors to simplify the equation

$y = (x + 8 - 5 i) (x + 8 + 5 i) (x - 7 i) (x + 7 i)$ .

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Step 7.1

Expand

$(x + 8 - 5 i) (x + 8 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot 8 + x (5 i) + 8 x + 8 \cdot 8 + 8 (5 i) - 5 i x - 5 i \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2

Simplify terms.

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Step 7.2.1

Combine the opposite terms in

$x \cdot x + x \cdot 8 + x (5 i) + 8 x + 8 \cdot 8 + 8 (5 i) - 5 i x - 5 i \cdot 8 - 5 i (5 i)$ .

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Step 7.2.1.1

Reorder the factors in the terms

$x (5 i)$ and

$- 5 i x$ .

$y = (x \cdot x + x \cdot 8 + 5 i x + 8 x + 8 \cdot 8 + 8 (5 i) - 5 i x - 5 i \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.1.2

Subtract

$5 i x$ from

$5 i x$ .

$y = (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 + 8 (5 i) + 0 - 5 i \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.1.3

Add

$x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 + 8 (5 i)$ and

$0$ .

$y = (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 + 8 (5 i) - 5 i \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.1.4

Reorder the factors in the terms

$8 (5 i)$ and

$- 5 i \cdot 8$ .

$y = (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 + 5 \cdot (8 i) - 5 \cdot (8 i) - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.1.5

Subtract

$5 \cdot 8 i$ from

$5 \cdot 8 i$ .

$y = (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 + 0 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.1.6

Add

$x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8$ and

$0$ .

$y = (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.2

Simplify each term.

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Step 7.2.2.1

Multiply

$x$ by

$x$ .

$y = (x^{2} + x \cdot 8 + 8 x + 8 \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.2.2

Move

$8$ to the left of

$x$ .

$y = (x^{2} + 8 \cdot x + 8 x + 8 \cdot 8 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.2.3

Multiply

$8$ by

$8$ .

$y = (x^{2} + 8 x + 8 x + 64 - 5 i (5 i)) (x - 7 i) (x + 7 i)$

Step 7.2.2.4

Multiply

$- 5 i (5 i)$ .

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Step 7.2.2.4.1

Multiply

$5$ by

$- 5$ .

$y = (x^{2} + 8 x + 8 x + 64 - 25 i i) (x - 7 i) (x + 7 i)$

Step 7.2.2.4.2

Raise

$i$ to the power of

$1$ .

$y = (x^{2} + 8 x + 8 x + 64 - 25 (i i)) (x - 7 i) (x + 7 i)$

Step 7.2.2.4.3

Raise

$i$ to the power of

$1$ .

$y = (x^{2} + 8 x + 8 x + 64 - 25 (i i)) (x - 7 i) (x + 7 i)$

Step 7.2.2.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = (x^{2} + 8 x + 8 x + 64 - 25 i^{1 + 1}) (x - 7 i) (x + 7 i)$

Step 7.2.2.4.5

Add

$1$ and

$1$ .

$y = (x^{2} + 8 x + 8 x + 64 - 25 i^{2}) (x - 7 i) (x + 7 i)$

Step 7.2.2.5

Rewrite

$i^{2}$ as

$- 1$ .

$y = (x^{2} + 8 x + 8 x + 64 - 25 \cdot - 1) (x - 7 i) (x + 7 i)$

Step 7.2.2.6

Multiply

$- 25$ by

$- 1$ .

$y = (x^{2} + 8 x + 8 x + 64 + 25) (x - 7 i) (x + 7 i)$

Step 7.2.3

Simplify by adding terms.

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Step 7.2.3.1

Add

$8 x$ and

$8 x$ .

$y = (x^{2} + 16 x + 64 + 25) (x - 7 i) (x + 7 i)$

Step 7.2.3.2

Add

$64$ and

$25$ .

$y = (x^{2} + 16 x + 89) (x - 7 i) (x + 7 i)$

Step 7.3

Expand

$(x^{2} + 16 x + 89) (x - 7 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} (- 7 i) + 16 x \cdot x + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4

Simplify each term.

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Step 7.4.1

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 7.4.1.1

Multiply

$x^{2}$ by

$x$ .

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Step 7.4.1.1.1

Raise

$x$ to the power of

$1$ .

$y = (x^{2} x + x^{2} (- 7 i) + 16 x \cdot x + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = (x^{2 + 1} + x^{2} (- 7 i) + 16 x \cdot x + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.1.2

Add

$2$ and

$1$ .

$y = (x^{3} + x^{2} (- 7 i) + 16 x \cdot x + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.2

Multiply

$x$ by

$x$ by adding the exponents.

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Step 7.4.2.1

Move

$x$ .

$y = (x^{3} + x^{2} (- 7 i) + 16 (x \cdot x) + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.2.2

Multiply

$x$ by

$x$ .

$y = (x^{3} + x^{2} (- 7 i) + 16 x^{2} + 16 x (- 7 i) + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.3

Multiply

$- 7$ by

$16$ .

$y = (x^{3} + x^{2} (- 7 i) + 16 x^{2} - 112 x i + 89 x + 89 (- 7 i)) (x + 7 i)$

Step 7.4.4

Multiply

$- 7$ by

$89$ .

$y = (x^{3} + x^{2} (- 7 i) + 16 x^{2} - 112 x i + 89 x - 623 i) (x + 7 i)$

Step 7.5

Expand

$(x^{3} + x^{2} (- 7 i) + 16 x^{2} - 112 x i + 89 x - 623 i) (x + 7 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (7 i) + x^{2} (- 7 i) x + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6

Simplify terms.

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Step 7.6.1

Simplify each term.

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Step 7.6.1.1

Multiply

$x^{3}$ by

$x$ by adding the exponents.

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Step 7.6.1.1.1

Multiply

$x^{3}$ by

$x$ .

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Step 7.6.1.1.1.1

Raise

$x$ to the power of

$1$ .

$y = x^{3} x + x^{3} (7 i) + x^{2} (- 7 i) x + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{3 + 1} + x^{3} (7 i) + x^{2} (- 7 i) x + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.1.2

Add

$3$ and

$1$ .

$y = x^{4} + x^{3} (7 i) + x^{2} (- 7 i) x + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.2

Move

$7$ to the left of

$x^{3}$ .

$y = x^{4} + 7 \cdot (x^{3} i) + x^{2} (- 7 i) x + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.3

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 7.6.1.3.1

Move

$x$ .

$y = x^{4} + 7 x^{3} i + x \cdot x^{2} (- 7 i) + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.3.2

Multiply

$x$ by

$x^{2}$ .

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Step 7.6.1.3.2.1

Raise

$x$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x \cdot x^{2} (- 7 i) + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{4} + 7 x^{3} i + x^{1 + 2} (- 7 i) + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.3.3

Add

$1$ and

$2$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 7 i) (7 i) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.4

Multiply

$x^{2} (- 7 i) (7 i)$ .

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Step 7.6.1.4.1

Multiply

$7$ by

$- 7$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 i) i + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.4.2

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 (i i)) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.4.3

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 (i i)) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 i^{1 + 1}) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.4.5

Add

$1$ and

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 i^{2}) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} (- 49 \cdot - 1) + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.6

Multiply

$- 49$ by

$- 1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + x^{2} \cdot 49 + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.7

Move

$49$ to the left of

$x^{2}$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 \cdot x^{2} + 16 x^{2} x + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.8

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 7.6.1.8.1

Move

$x$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 (x \cdot x^{2}) + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.8.2

Multiply

$x$ by

$x^{2}$ .

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Step 7.6.1.8.2.1

Raise

$x$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 (x \cdot x^{2}) + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.8.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{1 + 2} + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.8.3

Add

$1$ and

$2$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 16 x^{2} (7 i) - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.9

Multiply

$7$ by

$16$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x i x - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.10

Multiply

$x$ by

$x$ by adding the exponents.

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Step 7.6.1.10.1

Move

$x$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 (x \cdot x) i - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.10.2

Multiply

$x$ by

$x$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 112 x i (7 i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.11

Multiply

$- 112 x i (7 i)$ .

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Step 7.6.1.11.1

Multiply

$7$ by

$- 112$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x i i + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.11.2

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x (i i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.11.3

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x (i i) + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.11.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x i^{1 + 1} + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.11.5

Add

$1$ and

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x i^{2} + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.12

Rewrite

$i^{2}$ as

$- 1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i - 784 x \cdot - 1 + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.13

Multiply

$- 1$ by

$- 784$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x \cdot x + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.14

Multiply

$x$ by

$x$ by adding the exponents.

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Step 7.6.1.14.1

Move

$x$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 (x \cdot x) + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.14.2

Multiply

$x$ by

$x$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 89 x (7 i) - 623 i x - 623 i (7 i)$

Step 7.6.1.15

Multiply

$7$ by

$89$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 623 i (7 i)$

Step 7.6.1.16

Multiply

$- 623 i (7 i)$ .

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Step 7.6.1.16.1

Multiply

$7$ by

$- 623$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 i i$

Step 7.6.1.16.2

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 (i i)$

Step 7.6.1.16.3

Raise

$i$ to the power of

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 (i i)$

Step 7.6.1.16.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 i^{1 + 1}$

Step 7.6.1.16.5

Add

$1$ and

$1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 i^{2}$

Step 7.6.1.17

Rewrite

$i^{2}$ as

$- 1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x - 4361 \cdot - 1$

Step 7.6.1.18

Multiply

$- 4361$ by

$- 1$ .

$y = x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2

Simplify by adding terms.

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Step 7.6.2.1

Combine the opposite terms in

$x^{4} + 7 x^{3} i + x^{3} (- 7 i) + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$ .

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Step 7.6.2.1.1

Reorder the factors in the terms

$7 x^{3} i$ and

$x^{3} (- 7 i)$ .

$y = x^{4} + 7 i x^{3} - 7 i x^{3} + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2.1.2

Subtract

$7 i x^{3}$ from

$7 i x^{3}$ .

$y = x^{4} + 0 + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2.1.3

Add

$x^{4}$ and

$0$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 112 x^{2} i - 112 x^{2} i + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2.1.4

Subtract

$112 x^{2} i$ from

$112 x^{2} i$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 0 + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2.1.5

Add

$x^{4} + 49 x^{2} + 16 x^{3}$ and

$0$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 784 x + 89 x^{2} + 623 x i - 623 i x + 4361$

Step 7.6.2.1.6

Reorder the factors in the terms

$623 x i$ and

$- 623 i x$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 784 x + 89 x^{2} + 623 i x - 623 i x + 4361$

Step 7.6.2.1.7

Subtract

$623 i x$ from

$623 i x$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 784 x + 89 x^{2} + 0 + 4361$

Step 7.6.2.1.8

Add

$x^{4} + 49 x^{2} + 16 x^{3} + 784 x + 89 x^{2}$ and

$0$ .

$y = x^{4} + 49 x^{2} + 16 x^{3} + 784 x + 89 x^{2} + 4361$

Step 7.6.2.2

Add

$49 x^{2}$ and

$89 x^{2}$ .

$y = x^{4} + 138 x^{2} + 16 x^{3} + 784 x + 4361$

Step 7.6.2.3

Move

$138 x^{2}$ .

$y = x^{4} + 16 x^{3} + 138 x^{2} + 784 x + 4361$

Step 8