Find the Equation Given the Roots 5 , -5-6i

$5$ , $- 5 - 6 i$

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

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

The factor is $x - 5$

The root at $x = - 5 - 6 i$ was found by solving for $x$ when $x - (- 5 - 6 i) = y$ and $y = 0$ .

The factor is $x + 5 + 6 i$

The root at $x = - 5 + 6 i$ was found by solving for $x$ when $x - (- 5 + 6 i) = y$ and $y = 0$ .

The factor is $x + 5 - 6 i$

Combine all the factors into a single equation.

$y = (x - 5) (x + 5 + 6 i) (x + 5 - 6 i)$

Multiply all the factors to simplify the equation $y = (x - 5) (x + 5 + 6 i) (x + 5 - 6 i)$ .

Expand $(x - 5) (x + 5 + 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot 5 + x (6 i) - 5 x - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot 5 + x (6 i) - 5 x - 5 \cdot 5 - 5 (6 i)$ .

Reorder the factors in the terms $x \cdot 5$ and $- 5 x$ .

$y = (x \cdot x + 5 x + x (6 i) - 5 x - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Subtract $5 x$ from $5 x$ .

$y = (x \cdot x + x (6 i) + 0 - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Add $x \cdot x + x (6 i)$ and $0$ .

$y = (x \cdot x + x (6 i) - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x (6 i) - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Move $6$ to the left of $x$ .

$y = (x^{2} + 6 \cdot (x i) - 5 \cdot 5 - 5 (6 i)) (x + 5 - 6 i)$

Multiply $- 5$ by $5$ .

$y = (x^{2} + 6 x i - 25 - 5 (6 i)) (x + 5 - 6 i)$

Multiply $6$ by $- 5$ .

$y = (x^{2} + 6 x i - 25 - 30 i) (x + 5 - 6 i)$

Expand $(x^{2} + 6 x i - 25 - 30 i) (x + 5 - 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 5 + x^{2} (- 6 i) + 6 x i x + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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 $1$ .

$y = x^{2} x + x^{2} \cdot 5 + x^{2} (- 6 i) + 6 x i x + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

$y = x^{2 + 1} + x^{2} \cdot 5 + x^{2} (- 6 i) + 6 x i x + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot 5 + x^{2} (- 6 i) + 6 x i x + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Move $5$ to the left of $x^{2}$ .

$y = x^{3} + 5 \cdot x^{2} + x^{2} (- 6 i) + 6 x i x + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

Move $x$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 (x \cdot x) i + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 6 x i \cdot 5 + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $5$ by $6$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 6 x i (- 6 i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $6 x i (- 6 i)$ .

Multiply $- 6$ by $6$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x i i - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x (i i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x (i i) - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x i^{1 + 1} - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Add $1$ and $1$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x i^{2} - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i - 36 x \cdot - 1 - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $- 1$ by $- 36$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 25 \cdot 5 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $- 25$ by $5$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 - 25 (- 6 i) - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $- 6$ by $- 25$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 30 i \cdot 5 - 30 i (- 6 i)$

Multiply $5$ by $- 30$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 30 i (- 6 i)$

Multiply $- 30 i (- 6 i)$ .

Multiply $- 6$ by $- 30$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 i i$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 (i i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 (i i)$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 i^{2}$

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

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i + 180 \cdot - 1$

Multiply $180$ by $- 1$ .

$y = x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 5 x^{2} + x^{2} (- 6 i) + 6 x^{2} i + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$ .

Reorder the factors in the terms $x^{2} (- 6 i)$ and $6 x^{2} i$ .

$y = x^{3} + 5 x^{2} - 6 i x^{2} + 6 i x^{2} + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$

Add $- 6 i x^{2}$ and $6 i x^{2}$ .

$y = x^{3} + 5 x^{2} + 0 + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$

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

$y = x^{3} + 5 x^{2} + 30 x i + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$

Reorder the factors in the terms $30 x i$ and $- 30 i x$ .

$y = x^{3} + 5 x^{2} + 30 i x + 36 x - 25 x - 125 + 150 i - 30 i x - 150 i - 180$

Subtract $30 i x$ from $30 i x$ .

$y = x^{3} + 5 x^{2} + 36 x - 25 x - 125 + 150 i + 0 - 150 i - 180$

Add $x^{3} + 5 x^{2} + 36 x - 25 x - 125 + 150 i$ and $0$ .

$y = x^{3} + 5 x^{2} + 36 x - 25 x - 125 + 150 i - 150 i - 180$

Subtract $150 i$ from $150 i$ .

$y = x^{3} + 5 x^{2} + 36 x - 25 x - 125 + 0 - 180$

Add $x^{3} + 5 x^{2} + 36 x - 25 x - 125$ and $0$ .

$y = x^{3} + 5 x^{2} + 36 x - 25 x - 125 - 180$

Subtract $25 x$ from $36 x$ .

$y = x^{3} + 5 x^{2} + 11 x - 125 - 180$

Subtract $180$ from $- 125$ .

$y = x^{3} + 5 x^{2} + 11 x - 305$

Find the Equation Given the Roots 5 , -5-6i

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