Find the Equation Given the Roots -5 , 1+4i , -4i

$- 5$ , $1 + 4 i$ , $- 4 i$

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 $x = - 5$ was found by solving for $x$ when $x - (- 5) = y$ and $y = 0$ .

The factor is $x + 5$

Step 3

The root at $x = 1 + 4 i$ was found by solving for $x$ when $x - (1 + 4 i) = y$ and $y = 0$ .

The factor is $x - 1 - 4 i$

Step 4

The root at $x = 1 - 4 i$ was found by solving for $x$ when $x - (1 - 4 i) = y$ and $y = 0$ .

The factor is $x - 1 + 4 i$

Step 5

The root at $x = - 4 i$ was found by solving for $x$ when $x - (- 4 i) = y$ and $y = 0$ .

The factor is $x + 4 i$

Step 6

The root at $x = 4 i$ was found by solving for $x$ when $x - (4 i) = y$ and $y = 0$ .

The factor is $x - 4 i$

Step 7

Combine all the factors into a single equation.

$y = (x + 5) (x - 1 - 4 i) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Step 8

Multiply all the factors to simplify the equation $y = (x + 5) (x - 1 - 4 i) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$ .

Expand $(x + 5) (x - 1 - 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 1 + x (- 4 i) + 5 x + 5 \cdot - 1 + 5 (- 4 i)) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 1 + x (- 4 i) + 5 x + 5 \cdot - 1 + 5 (- 4 i)) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Move $- 1$ to the left of $x$ .

$y = (x^{2} - 1 \cdot x + x (- 4 i) + 5 x + 5 \cdot - 1 + 5 (- 4 i)) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Rewrite $- 1 x$ as $- x$ .

$y = (x^{2} - x + x (- 4 i) + 5 x + 5 \cdot - 1 + 5 (- 4 i)) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Multiply $5$ by $- 1$ .

$y = (x^{2} - x + x (- 4 i) + 5 x - 5 + 5 (- 4 i)) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Multiply $- 4$ by $5$ .

$y = (x^{2} - x + x (- 4 i) + 5 x - 5 - 20 i) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Add $- x$ and $5 x$ .

$y = (x^{2} + x (- 4 i) + 4 x - 5 - 20 i) (x - 1 + 4 i) (x + 4 i) (x - 4 i)$

Expand $(x^{2} + x (- 4 i) + 4 x - 5 - 20 i) (x - 1 + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 1 + x^{2} (4 i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Simplify terms.

Simplify each term.

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

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

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

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

$y = (x^{2 + 1} + x^{2} \cdot - 1 + x^{2} (4 i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 1 + x^{2} (4 i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Move $- 1$ to the left of $x^{2}$ .

$y = (x^{3} - 1 \cdot x^{2} + x^{2} (4 i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + x^{2} (4 i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 \cdot (x^{2} i) + x (- 4 i) x + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

Move $x$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x \cdot x (- 4 i) + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $x$ by $x$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + x (- 4 i) \cdot - 1 + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 1$ by $- 4$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + x (4 i) + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 \cdot (x i) + x (- 4 i) (4 i) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

Multiply $4$ by $- 4$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x (- 16 i) i + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x (- 16 (i i)) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x (- 16 i^{1 + 1}) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Add $1$ and $1$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x (- 16 i^{2}) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x (- 16 \cdot - 1) + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 16$ by $- 1$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + x \cdot 16 + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Move $16$ to the left of $x$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 \cdot x + 4 x \cdot x + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

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

Move $x$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 (x \cdot x) + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $x$ by $x$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} + 4 x \cdot - 1 + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 1$ by $4$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 4 x (4 i) - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $4$ by $4$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x - 5 \cdot - 1 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 5$ by $- 1$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 5 (4 i) - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $4$ by $- 5$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x - 20 i \cdot - 1 - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 1$ by $- 20$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 20 i (4 i)) (x + 4 i) (x - 4 i)$

Multiply $- 20 i (4 i)$ .

Multiply $4$ by $- 20$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 i i) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 (i i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 (i i)) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 i^{1 + 1}) (x + 4 i) (x - 4 i)$

Add $1$ and $1$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 i^{2}) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i - 80 \cdot - 1) (x + 4 i) (x - 4 i)$

Multiply $- 80$ by $- 1$ .

$y = (x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i + 80) (x + 4 i) (x - 4 i)$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - x^{2} + 4 x^{2} i + x^{2} (- 4 i) + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i + 80$ .

Reorder the factors in the terms $4 x^{2} i$ and $x^{2} (- 4 i)$ .

$y = (x^{3} - x^{2} + 4 i x^{2} - 4 i x^{2} + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i + 80) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 0 + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i + 80) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} - x^{2} + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i - 20 i x + 20 i + 80) (x + 4 i) (x - 4 i)$

Add $- 20 i$ and $20 i$ .

$y = (x^{3} - x^{2} + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i x + 0 + 80) (x + 4 i) (x - 4 i)$

Add $x^{3} - x^{2} + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i x$ and $0$ .

$y = (x^{3} - x^{2} + 4 x i + 16 x + 4 x^{2} - 4 x + 16 x i - 5 x + 5 - 20 i x + 80) (x + 4 i) (x - 4 i)$

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

$y = (x^{3} + 3 x^{2} + 4 x i + 16 x - 4 x + 16 x i - 5 x + 5 - 20 i x + 80) (x + 4 i) (x - 4 i)$

Add $4 x i$ and $16 x i$ .

$y = (x^{3} + 3 x^{2} + 20 x i + 16 x - 4 x - 5 x + 5 - 20 i x + 80) (x + 4 i) (x - 4 i)$

Combine the opposite terms in $x^{3} + 3 x^{2} + 20 x i + 16 x - 4 x - 5 x + 5 - 20 i x + 80$ .

Reorder the factors in the terms $20 x i$ and $- 20 i x$ .

$y = (x^{3} + 3 x^{2} + 20 i x + 16 x - 4 x - 5 x + 5 - 20 i x + 80) (x + 4 i) (x - 4 i)$

Subtract $20 i x$ from $20 i x$ .

$y = (x^{3} + 3 x^{2} + 16 x - 4 x - 5 x + 5 + 0 + 80) (x + 4 i) (x - 4 i)$

Add $x^{3} + 3 x^{2} + 16 x - 4 x - 5 x + 5$ and $0$ .

$y = (x^{3} + 3 x^{2} + 16 x - 4 x - 5 x + 5 + 80) (x + 4 i) (x - 4 i)$

Subtract $4 x$ from $16 x$ .

$y = (x^{3} + 3 x^{2} + 12 x - 5 x + 5 + 80) (x + 4 i) (x - 4 i)$

Subtract $5 x$ from $12 x$ .

$y = (x^{3} + 3 x^{2} + 7 x + 5 + 80) (x + 4 i) (x - 4 i)$

Add $5$ and $80$ .

$y = (x^{3} + 3 x^{2} + 7 x + 85) (x + 4 i) (x - 4 i)$

Expand $(x^{3} + 3 x^{2} + 7 x + 85) (x + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{3} x + x^{3} (4 i) + 3 x^{2} x + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Simplify each term.

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

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

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

$y = (x^{3} x + x^{3} (4 i) + 3 x^{2} x + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

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

$y = (x^{3 + 1} + x^{3} (4 i) + 3 x^{2} x + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Add $3$ and $1$ .

$y = (x^{4} + x^{3} (4 i) + 3 x^{2} x + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Move $4$ to the left of $x^{3}$ .

$y = (x^{4} + 4 \cdot (x^{3} i) + 3 x^{2} x + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

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

Move $x$ .

$y = (x^{4} + 4 x^{3} i + 3 (x \cdot x^{2}) + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

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

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

$y = (x^{4} + 4 x^{3} i + 3 (x \cdot x^{2}) + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

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

$y = (x^{4} + 4 x^{3} i + 3 x^{1 + 2} + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Add $1$ and $2$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 3 x^{2} (4 i) + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Multiply $4$ by $3$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 x \cdot x + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

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

Move $x$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 (x \cdot x) + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Multiply $x$ by $x$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 x^{2} + 7 x (4 i) + 85 x + 85 (4 i)) (x - 4 i)$

Multiply $4$ by $7$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 x^{2} + 28 x i + 85 x + 85 (4 i)) (x - 4 i)$

Multiply $4$ by $85$ .

$y = (x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 x^{2} + 28 x i + 85 x + 340 i) (x - 4 i)$

Expand $(x^{4} + 4 x^{3} i + 3 x^{3} + 12 x^{2} i + 7 x^{2} + 28 x i + 85 x + 340 i) (x - 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{4} x + x^{4} (- 4 i) + 4 x^{3} i x + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Simplify terms.

Simplify each term.

Multiply $x^{4}$ by $x$ by adding the exponents.

Multiply $x^{4}$ by $x$ .

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

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

$y = x^{4 + 1} + x^{4} (- 4 i) + 4 x^{3} i x + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $4$ and $1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{3} i x + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 (x \cdot x^{3}) i + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{1 + 3} i + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $3$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 4 x^{3} i (- 4 i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $4 x^{3} i (- 4 i)$ .

Multiply $- 4$ by $4$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i - 16 x^{3} i i + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i - 16 x^{3} (i i) + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i - 16 x^{3} i^{1 + 1} + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i - 16 x^{3} i^{2} + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i - 16 x^{3} \cdot - 1 + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 1$ by $- 16$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{3} x + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 (x \cdot x^{3}) + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{1 + 3} + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $3$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} + 3 x^{3} (- 4 i) + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 4$ by $3$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{2} i x + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 (x \cdot x^{2}) i + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{1 + 2} i + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $2$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 12 x^{2} i (- 4 i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $12 x^{2} i (- 4 i)$ .

Multiply $- 4$ by $12$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i - 48 x^{2} i i + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i - 48 x^{2} (i i) + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i - 48 x^{2} i^{1 + 1} + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i - 48 x^{2} i^{2} + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i - 48 x^{2} \cdot - 1 + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 1$ by $- 48$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{2} x + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 (x \cdot x^{2}) + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{1 + 2} + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $2$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} + 7 x^{2} (- 4 i) + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 4$ by $7$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x i x + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 (x \cdot x) i + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $x$ by $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 28 x i (- 4 i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $28 x i (- 4 i)$ .

Multiply $- 4$ by $28$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i - 112 x i i + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i - 112 x (i i) + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i - 112 x i^{1 + 1} + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Add $1$ and $1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i - 112 x i^{2} + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i - 112 x \cdot - 1 + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 1$ by $- 112$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x \cdot x + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

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

Move $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 (x \cdot x) + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $x$ by $x$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} + 85 x (- 4 i) + 340 i x + 340 i (- 4 i)$

Multiply $- 4$ by $85$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 340 i (- 4 i)$

Multiply $340 i (- 4 i)$ .

Multiply $- 4$ by $340$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 i i$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 (i i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 (i i)$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 i^{2}$

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

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x - 1360 \cdot - 1$

Multiply $- 1360$ by $- 1$ .

$y = x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Simplify by adding terms.

Combine the opposite terms in $x^{5} + x^{4} (- 4 i) + 4 x^{4} i + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$ .

Reorder the factors in the terms $x^{4} (- 4 i)$ and $4 x^{4} i$ .

$y = x^{5} - 4 i x^{4} + 4 i x^{4} + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $- 4 i x^{4}$ and $4 i x^{4}$ .

$y = x^{5} + 0 + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $x^{5}$ and $0$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} - 12 x^{3} i + 12 x^{3} i + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $- 12 x^{3} i$ and $12 x^{3} i$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 0 + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $x^{5} + 16 x^{3} + 3 x^{4}$ and $0$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} - 28 x^{2} i + 28 x^{2} i + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $- 28 x^{2} i$ and $28 x^{2} i$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 0 + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Add $x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3}$ and $0$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 112 x + 85 x^{2} - 340 x i + 340 i x + 1360$

Reorder the factors in the terms $- 340 x i$ and $340 i x$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 112 x + 85 x^{2} - 340 i x + 340 i x + 1360$

Add $- 340 i x$ and $340 i x$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 112 x + 85 x^{2} + 0 + 1360$

Add $x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 112 x + 85 x^{2}$ and $0$ .

$y = x^{5} + 16 x^{3} + 3 x^{4} + 48 x^{2} + 7 x^{3} + 112 x + 85 x^{2} + 1360$

Add $16 x^{3}$ and $7 x^{3}$ .

$y = x^{5} + 23 x^{3} + 3 x^{4} + 48 x^{2} + 112 x + 85 x^{2} + 1360$

Add $48 x^{2}$ and $85 x^{2}$ .

$y = x^{5} + 23 x^{3} + 3 x^{4} + 133 x^{2} + 112 x + 1360$

Move $23 x^{3}$ .

$y = x^{5} + 3 x^{4} + 23 x^{3} + 133 x^{2} + 112 x + 1360$

Step 9

Find the Equation Given the Roots -5 , 1+4i , -4i

Related Questions

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