Find the Equation Given the Roots 2 , -3 , 1-i

$2$ , $- 3$ , $1 - i$

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

The factor is $x + 3$

Step 4

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

The factor is $x - 1 + i$

Step 5

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

The factor is $x - 1 - i$

Step 6

Combine all the factors into a single equation.

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

Step 7

Multiply all the factors to simplify the equation $y = (x - 2) (x + 3) (x - 1 + i) (x - 1 - i)$ .

Expand $(x - 2) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$y = (x \cdot x + x \cdot 3 - 2 (x + 3)) (x - 1 + i) (x - 1 - i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 3 - 2 x - 2 \cdot 3) (x - 1 + i) (x - 1 - i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Multiply $- 2$ by $3$ .

$y = (x^{2} + 3 x - 2 x - 6) (x - 1 + i) (x - 1 - i)$

Subtract $2 x$ from $3 x$ .

$y = (x^{2} + x - 6) (x - 1 + i) (x - 1 - i)$

Expand $(x^{2} + x - 6) (x - 1 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 1 + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot - 1 + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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

$y = (x^{2 + 1} + x^{2} \cdot - 1 + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 1 + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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

$y = (x^{3} - 1 \cdot x^{2} + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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

$y = (x^{3} - x^{2} + x^{2} i + x \cdot x + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

Multiply $x$ by $x$ .

$y = (x^{3} - x^{2} + x^{2} i + x^{2} + x \cdot - 1 + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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

$y = (x^{3} - x^{2} + x^{2} i + x^{2} - 1 \cdot x + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

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

$y = (x^{3} - x^{2} + x^{2} i + x^{2} - x + x i - 6 x - 6 \cdot - 1 - 6 i) (x - 1 - i)$

Multiply $- 6$ by $- 1$ .

$y = (x^{3} - x^{2} + x^{2} i + x^{2} - x + x i - 6 x + 6 - 6 i) (x - 1 - i)$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - x^{2} + x^{2} i + x^{2} - x + x i - 6 x + 6 - 6 i$ .

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

$y = (x^{3} + x^{2} i + 0 - x + x i - 6 x + 6 - 6 i) (x - 1 - i)$

Add $x^{3} + x^{2} i$ and $0$ .

$y = (x^{3} + x^{2} i - x + x i - 6 x + 6 - 6 i) (x - 1 - i)$

Subtract $6 x$ from $- x$ .

$y = (x^{3} + x^{2} i + x i - 7 x + 6 - 6 i) (x - 1 - i)$

Expand $(x^{3} + x^{2} i + x i - 7 x + 6 - 6 i) (x - 1 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 1 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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.

$y = x^{3 + 1} + x^{3} \cdot - 1 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 1 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Move $- 1$ to the left of $x^{3}$ .

$y = x^{4} - 1 \cdot x^{3} + x^{3} (- i) + x^{2} i x + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{2} i x + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

Move $x$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x \cdot x^{2} i + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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.

$y = x^{4} - x^{3} + x^{3} (- i) + x^{1 + 2} i + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Add $1$ and $2$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i + x^{2} i \cdot - 1 + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - 1 \cdot (x^{2} i) + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Rewrite $- 1 (x^{2} i)$ as $- (x^{2} i)$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - (x^{2} i) + x^{2} i (- i) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} (- (i i)) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} (- i^{1 + 1}) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Add $1$ and $1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} (- i^{2}) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} (1) + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} \cdot 1 + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $x^{2}$ by $1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x i x + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

Move $x$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x \cdot x i + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i + x i \cdot - 1 + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - 1 \cdot (x i) + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Rewrite $- 1 (x i)$ as $- (x i)$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - (x i) + x i (- i) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $x i (- i)$ .

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x (- i^{1 + 1}) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Add $1$ and $1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x (- i^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x (1) - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x \cdot 1 - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $x$ by $1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x \cdot x - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

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

Move $x$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 (x \cdot x) - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} - 7 x \cdot - 1 - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $- 7$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x - 7 x (- i) + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $- 7$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x + 6 \cdot - 1 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $6$ by $- 1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 + 6 (- i) - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $6$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x - 6 i \cdot - 1 - 6 i (- i)$

Multiply $- 1$ by $- 6$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6 i (- i)$

Multiply $- 6 i (- i)$ .

Multiply $- 1$ by $- 6$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 i i$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 (i i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 (i i)$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 i^{2}$

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

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i + 6 \cdot - 1$

Multiply $6$ by $- 1$ .

$y = x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

Simplify by adding terms.

Combine the opposite terms in $x^{4} - x^{3} + x^{3} (- i) + x^{3} i - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$ .

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

$y = x^{4} - x^{3} - i x^{3} + i x^{3} - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

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

$y = x^{4} - x^{3} + 0 - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

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

$y = x^{4} - x^{3} - x^{2} i + x^{2} + x^{2} i - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

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

$y = x^{4} - x^{3} + x^{2} + 0 - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

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

$y = x^{4} - x^{3} + x^{2} - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i - 6 i x + 6 i - 6$

Add $- 6 i$ and $6 i$ .

$y = x^{4} - x^{3} + x^{2} - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i x + 0 - 6$

Add $x^{4} - x^{3} + x^{2} - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i x$ and $0$ .

$y = x^{4} - x^{3} + x^{2} - x i + x - 7 x^{2} + 7 x + 7 x i + 6 x - 6 - 6 i x - 6$

Subtract $7 x^{2}$ from $x^{2}$ .

$y = x^{4} - x^{3} - 6 x^{2} - x i + x + 7 x + 7 x i + 6 x - 6 - 6 i x - 6$

Add $- x i$ and $7 x i$ .

$y = x^{4} - x^{3} - 6 x^{2} + 6 x i + x + 7 x + 6 x - 6 - 6 i x - 6$

Combine the opposite terms in $x^{4} - x^{3} - 6 x^{2} + 6 x i + x + 7 x + 6 x - 6 - 6 i x - 6$ .

Reorder the factors in the terms $6 x i$ and $- 6 i x$ .

$y = x^{4} - x^{3} - 6 x^{2} + 6 i x + x + 7 x + 6 x - 6 - 6 i x - 6$

Subtract $6 i x$ from $6 i x$ .

$y = x^{4} - x^{3} - 6 x^{2} + x + 7 x + 6 x - 6 + 0 - 6$

Add $x^{4} - x^{3} - 6 x^{2} + x + 7 x + 6 x - 6$ and $0$ .

$y = x^{4} - x^{3} - 6 x^{2} + x + 7 x + 6 x - 6 - 6$

Add $x$ and $7 x$ .

$y = x^{4} - x^{3} - 6 x^{2} + 8 x + 6 x - 6 - 6$

Add $8 x$ and $6 x$ .

$y = x^{4} - x^{3} - 6 x^{2} + 14 x - 6 - 6$

Subtract $6$ from $- 6$ .

$y = x^{4} - x^{3} - 6 x^{2} + 14 x - 12$

Step 8

Find the Equation Given the Roots 2 , -3 , 1-i

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