Find the Equation Given the Roots 2 , 8 , 2-3i , 2+3i

$2$ , $8$ , $2 - 3 i$ , $2 + 3 i$

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

The factor is $x - 2$

Step 3

The root at $x = 8$ was found by solving for $x$ when $x - (8) = y$ and $y = 0$ .

The factor is $x - 8$

Step 4

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

The factor is $x - 2 + 3 i$

Step 5

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

The factor is $x - 2 - 3 i$

Step 6

Combine all the factors into a single equation.

$y = (x - 2) (x - 8) (x - 2 + 3 i) (x - 2 - 3 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x - 2) (x - 8) (x - 2 + 3 i) (x - 2 - 3 i)$ .

Expand $(x - 2) (x - 8)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 8) - 2 (x - 8)) (x - 2 + 3 i) (x - 2 - 3 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 8 - 2 (x - 8)) (x - 2 + 3 i) (x - 2 - 3 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 8 - 2 x - 2 \cdot - 8) (x - 2 + 3 i) (x - 2 - 3 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 8 - 2 x - 2 \cdot - 8) (x - 2 + 3 i) (x - 2 - 3 i)$

Move $- 8$ to the left of $x$ .

$y = (x^{2} - 8 \cdot x - 2 x - 2 \cdot - 8) (x - 2 + 3 i) (x - 2 - 3 i)$

Multiply $- 2$ by $- 8$ .

$y = (x^{2} - 8 x - 2 x + 16) (x - 2 + 3 i) (x - 2 - 3 i)$

Subtract $2 x$ from $- 8 x$ .

$y = (x^{2} - 10 x + 16) (x - 2 + 3 i) (x - 2 - 3 i)$

Expand $(x^{2} - 10 x + 16) (x - 2 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 2 + x^{2} (3 i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot - 2 + x^{2} (3 i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

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

$y = (x^{2 + 1} + x^{2} \cdot - 2 + x^{2} (3 i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 2 + x^{2} (3 i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

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

$y = (x^{3} - 2 \cdot x^{2} + x^{2} (3 i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

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

$y = (x^{3} - 2 x^{2} + 3 \cdot (x^{2} i) - 10 x \cdot x - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

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

Move $x$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 (x \cdot x) - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

Multiply $x$ by $x$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 x^{2} - 10 x \cdot - 2 - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

Multiply $- 2$ by $- 10$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 x^{2} + 20 x - 10 x (3 i) + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

Multiply $3$ by $- 10$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 x^{2} + 20 x - 30 x i + 16 x + 16 \cdot - 2 + 16 (3 i)) (x - 2 - 3 i)$

Multiply $16$ by $- 2$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 x^{2} + 20 x - 30 x i + 16 x - 32 + 16 (3 i)) (x - 2 - 3 i)$

Multiply $3$ by $16$ .

$y = (x^{3} - 2 x^{2} + 3 x^{2} i - 10 x^{2} + 20 x - 30 x i + 16 x - 32 + 48 i) (x - 2 - 3 i)$

Simplify by adding terms.

Subtract $10 x^{2}$ from $- 2 x^{2}$ .

$y = (x^{3} - 12 x^{2} + 3 x^{2} i + 20 x - 30 x i + 16 x - 32 + 48 i) (x - 2 - 3 i)$

Add $20 x$ and $16 x$ .

$y = (x^{3} - 12 x^{2} + 3 x^{2} i - 30 x i + 36 x - 32 + 48 i) (x - 2 - 3 i)$

Expand $(x^{3} - 12 x^{2} + 3 x^{2} i - 30 x i + 36 x - 32 + 48 i) (x - 2 - 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 2 + x^{3} (- 3 i) - 12 x^{2} x - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Simplify terms.

Simplify each term.

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

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

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

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

$y = x^{3 + 1} + x^{3} \cdot - 2 + x^{3} (- 3 i) - 12 x^{2} x - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 2 + x^{3} (- 3 i) - 12 x^{2} x - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

$y = x^{4} - 2 \cdot x^{3} + x^{3} (- 3 i) - 12 x^{2} x - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 (x \cdot x^{2}) - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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.

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{1 + 2} - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} - 12 x^{2} \cdot - 2 - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 2$ by $- 12$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} - 12 x^{2} (- 3 i) + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 3$ by $- 12$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{2} i x + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 (x \cdot x^{2}) i + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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.

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{1 + 2} i + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i + 3 x^{2} i \cdot - 2 + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 2$ by $3$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 3 x^{2} i (- 3 i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

Multiply $- 3$ by $3$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i - 9 x^{2} i i - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i - 9 x^{2} (i i) - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i - 9 x^{2} i^{1 + 1} - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i - 9 x^{2} i^{2} - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i - 9 x^{2} \cdot - 1 - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 1$ by $- 9$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x i x - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 (x \cdot x) i - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i - 30 x i \cdot - 2 - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 2$ by $- 30$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 30 x i (- 3 i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 30 x i (- 3 i)$ .

Multiply $- 3$ by $- 30$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i + 90 x i i + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i + 90 x (i i) + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i + 90 x i^{1 + 1} + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i + 90 x i^{2} + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i + 90 x \cdot - 1 + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 1$ by $90$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x \cdot x + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

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

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 (x \cdot x) + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} + 36 x \cdot - 2 + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 2$ by $36$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x + 36 x (- 3 i) - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 3$ by $36$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x - 32 \cdot - 2 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 32$ by $- 2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 - 32 (- 3 i) + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 3$ by $- 32$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x + 48 i \cdot - 2 + 48 i (- 3 i)$

Multiply $- 2$ by $48$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 48 i (- 3 i)$

Multiply $48 i (- 3 i)$ .

Multiply $- 3$ by $48$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i - 144 i i$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i - 144 (i i)$

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

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i - 144 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i - 144 i^{2}$

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

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i - 144 \cdot - 1$

Multiply $- 144$ by $- 1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 144$

Simplify by adding terms.

Combine the opposite terms in $x^{4} - 2 x^{3} + x^{3} (- 3 i) - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 x^{3} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 144$ .

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

$y = x^{4} - 2 x^{3} - 3 i x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 3 i x^{3} - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 144$

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

$y = x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i + 0 - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 144$

Add $x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i$ and $0$ .

$y = x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 96 i + 48 i x - 96 i + 144$

Subtract $96 i$ from $96 i$ .

$y = x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 48 i x + 0 + 144$

Add $x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 48 i x$ and $0$ .

$y = x^{4} - 2 x^{3} - 12 x^{3} + 24 x^{2} + 36 x^{2} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Subtract $12 x^{3}$ from $- 2 x^{3}$ .

$y = x^{4} - 14 x^{3} + 24 x^{2} + 36 x^{2} i - 6 x^{2} i + 9 x^{2} - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Add $24 x^{2}$ and $9 x^{2}$ .

$y = x^{4} - 14 x^{3} + 33 x^{2} + 36 x^{2} i - 6 x^{2} i - 30 x^{2} i + 60 x i - 90 x + 36 x^{2} - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Add $33 x^{2}$ and $36 x^{2}$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} + 36 x^{2} i - 6 x^{2} i - 30 x^{2} i + 60 x i - 90 x - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Subtract $6 x^{2} i$ from $36 x^{2} i$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} + 30 x^{2} i - 30 x^{2} i + 60 x i - 90 x - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Combine the opposite terms in $x^{4} - 14 x^{3} + 69 x^{2} + 30 x^{2} i - 30 x^{2} i + 60 x i - 90 x - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$ .

Subtract $30 x^{2} i$ from $30 x^{2} i$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} + 0 + 60 x i - 90 x - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Add $x^{4} - 14 x^{3} + 69 x^{2}$ and $0$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} + 60 x i - 90 x - 72 x - 108 x i - 32 x + 64 + 48 i x + 144$

Subtract $108 x i$ from $60 x i$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 48 x i - 90 x - 72 x - 32 x + 64 + 48 i x + 144$

Combine the opposite terms in $x^{4} - 14 x^{3} + 69 x^{2} - 48 x i - 90 x - 72 x - 32 x + 64 + 48 i x + 144$ .

Reorder the factors in the terms $- 48 x i$ and $48 i x$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 48 i x - 90 x - 72 x - 32 x + 64 + 48 i x + 144$

Add $- 48 i x$ and $48 i x$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 90 x - 72 x - 32 x + 64 + 0 + 144$

Add $x^{4} - 14 x^{3} + 69 x^{2} - 90 x - 72 x - 32 x + 64$ and $0$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 90 x - 72 x - 32 x + 64 + 144$

Subtract $72 x$ from $- 90 x$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 162 x - 32 x + 64 + 144$

Subtract $32 x$ from $- 162 x$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 194 x + 64 + 144$

Add $64$ and $144$ .

$y = x^{4} - 14 x^{3} + 69 x^{2} - 194 x + 208$

Step 8

Find the Equation Given the Roots 2 , 8 , 2-3i , 2+3i

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