Find the Equation Given the Roots 0 , 3i , 3+i

$0$ , $3 i$ , $3 + i$

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

The factor is $x$

Step 3

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

The factor is $x - 3 i$

Step 4

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

The factor is $x + 3 i$

Step 5

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

The factor is $x - 3 - i$

Step 6

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

The factor is $x - 3 + i$

Step 7

Combine all the factors into a single equation.

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

Step 8

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

Simplify by multiplying through.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Expand $(x^{2} + x (- 3 i)) (x + 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x (- 3 i) (3 i)$ .

Multiply $3$ by $- 3$ .

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 i) i) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 (i i))) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 (i i))) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 i^{1 + 1})) (x - 3 - i) (x - 3 + i)$

Add $1$ and $1$ .

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 i^{2})) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x (- 9 \cdot - 1)) (x - 3 - i) (x - 3 + i)$

Multiply $- 9$ by $- 1$ .

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + x \cdot 9) (x - 3 - i) (x - 3 + i)$

Move $9$ to the left of $x$ .

$y = (x^{3} + 3 x^{2} i + x^{2} (- 3 i) + 9 x) (x - 3 - i) (x - 3 + i)$

Add $3 x^{2} i$ and $x^{2} (- 3 i)$ .

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

$y = (x^{3} + 3 x^{2} i - 3 \cdot (x^{2} i) + 9 x) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 0 + 9 x) (x - 3 - i) (x - 3 + i)$

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

$y = (x^{3} + 9 x) (x - 3 - i) (x - 3 + i)$

Expand $(x^{3} + 9 x) (x - 3 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{3} x + x^{3} \cdot - 3 + x^{3} (- i) + 9 x \cdot x + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

Simplify each term.

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

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

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

$y = (x^{3} x + x^{3} \cdot - 3 + x^{3} (- i) + 9 x \cdot x + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

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

$y = (x^{3 + 1} + x^{3} \cdot - 3 + x^{3} (- i) + 9 x \cdot x + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

Add $3$ and $1$ .

$y = (x^{4} + x^{3} \cdot - 3 + x^{3} (- i) + 9 x \cdot x + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

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

$y = (x^{4} - 3 \cdot x^{3} + x^{3} (- i) + 9 x \cdot x + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

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

Move $x$ .

$y = (x^{4} - 3 x^{3} + x^{3} (- i) + 9 (x \cdot x) + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

Multiply $x$ by $x$ .

$y = (x^{4} - 3 x^{3} + x^{3} (- i) + 9 x^{2} + 9 x \cdot - 3 + 9 x (- i)) (x - 3 + i)$

Multiply $- 3$ by $9$ .

$y = (x^{4} - 3 x^{3} + x^{3} (- i) + 9 x^{2} - 27 x + 9 x (- i)) (x - 3 + i)$

Multiply $- 1$ by $9$ .

$y = (x^{4} - 3 x^{3} + x^{3} (- i) + 9 x^{2} - 27 x - 9 x i) (x - 3 + i)$

Expand $(x^{4} - 3 x^{3} + x^{3} (- i) + 9 x^{2} - 27 x - 9 x i) (x - 3 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{4} x + x^{4} \cdot - 3 + x^{4} i - 3 x^{3} x - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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.

$y = x^{4 + 1} + x^{4} \cdot - 3 + x^{4} i - 3 x^{3} x - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Add $4$ and $1$ .

$y = x^{5} + x^{4} \cdot - 3 + x^{4} i - 3 x^{3} x - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

$y = x^{5} - 3 \cdot x^{4} + x^{4} i - 3 x^{3} x - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

Move $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 (x \cdot x^{3}) - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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.

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{1 + 3} - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Add $1$ and $3$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} - 3 x^{3} \cdot - 3 - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 3$ by $- 3$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{3} (- i) x + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

Move $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x \cdot x^{3} (- i) + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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.

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{1 + 3} (- i) + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Add $1$ and $3$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + x^{3} (- i) \cdot - 3 + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 3$ by $- 1$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + x^{3} (3 i) + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 \cdot (x^{3} i) + x^{3} (- i) i + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} (- (i i)) + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} (- i^{1 + 1}) + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Add $1$ and $1$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} (- i^{2}) + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} (1) + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 1$ by $- 1$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} \cdot 1 + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $x^{3}$ by $1$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{2} x + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

Move $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 (x \cdot x^{2}) + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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.

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{1 + 2} + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Add $1$ and $2$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} + 9 x^{2} \cdot - 3 + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 3$ by $9$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x \cdot x - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

Move $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 (x \cdot x) - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $x$ by $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} - 27 x \cdot - 3 - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 3$ by $- 27$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x i x - 9 x i \cdot - 3 - 9 x i i$

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

Move $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 (x \cdot x) i - 9 x i \cdot - 3 - 9 x i i$

Multiply $x$ by $x$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i - 9 x i \cdot - 3 - 9 x i i$

Multiply $- 3$ by $- 9$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x i i$

Multiply $- 9 x i i$ .

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x (i i)$

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x (i i)$

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x i^{2}$

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

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i - 9 x \cdot - 1$

Multiply $- 1$ by $- 9$ .

$y = x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

Simplify by adding terms.

Combine the opposite terms in $x^{5} - 3 x^{4} + x^{4} i - 3 x^{4} + 9 x^{3} - 3 x^{3} i + x^{4} (- i) + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$ .

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

$y = x^{5} - 3 x^{4} + i x^{4} - 3 x^{4} + 9 x^{3} - 3 x^{3} i - i x^{4} + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

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

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} - 3 x^{3} i + 0 + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

Add $x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} - 3 x^{3} i$ and $0$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} - 3 x^{3} i + 3 x^{3} i + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

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

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + 0 + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

Add $x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3}$ and $0$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} + 9 x^{2} i - 27 x^{2} + 81 x - 27 x i - 9 x^{2} i + 27 x i + 9 x$

Subtract $9 x^{2} i$ from $9 x^{2} i$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x - 27 x i + 0 + 27 x i + 9 x$

Add $x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x - 27 x i$ and $0$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x - 27 x i + 27 x i + 9 x$

Add $- 27 x i$ and $27 x i$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x + 0 + 9 x$

Add $x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x$ and $0$ .

$y = x^{5} - 3 x^{4} - 3 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x + 9 x$

Subtract $3 x^{4}$ from $- 3 x^{4}$ .

$y = x^{5} - 6 x^{4} + 9 x^{3} + x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x + 9 x$

Add $9 x^{3}$ and $x^{3}$ .

$y = x^{5} - 6 x^{4} + 10 x^{3} + 9 x^{3} - 27 x^{2} - 27 x^{2} + 81 x + 9 x$

Add $10 x^{3}$ and $9 x^{3}$ .

$y = x^{5} - 6 x^{4} + 19 x^{3} - 27 x^{2} - 27 x^{2} + 81 x + 9 x$

Subtract $27 x^{2}$ from $- 27 x^{2}$ .

$y = x^{5} - 6 x^{4} + 19 x^{3} - 54 x^{2} + 81 x + 9 x$

Add $81 x$ and $9 x$ .

$y = x^{5} - 6 x^{4} + 19 x^{3} - 54 x^{2} + 90 x$

Step 9

Find the Equation Given the Roots 0 , 3i , 3+i

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