Find the Equation Given the Roots 4 , -14 , 5+8i

$4$ , $- 14$ , $5 + 8 i$

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

The factor is $x - 4$

Step 3

The root at $x = - 14$ was found by solving for $x$ when $x - (- 14) = y$ and $y = 0$ .

The factor is $x + 14$

Step 4

The root at $x = 5 + 8 i$ was found by solving for $x$ when $x - (5 + 8 i) = y$ and $y = 0$ .

The factor is $x - 5 - 8 i$

Step 5

The root at $x = 5 - 8 i$ was found by solving for $x$ when $x - (5 - 8 i) = y$ and $y = 0$ .

The factor is $x - 5 + 8 i$

Step 6

Combine all the factors into a single equation.

$y = (x - 4) (x + 14) (x - 5 - 8 i) (x - 5 + 8 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x - 4) (x + 14) (x - 5 - 8 i) (x - 5 + 8 i)$ .

Expand $(x - 4) (x + 14)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 14) - 4 (x + 14)) (x - 5 - 8 i) (x - 5 + 8 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 14 - 4 (x + 14)) (x - 5 - 8 i) (x - 5 + 8 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 14 - 4 x - 4 \cdot 14) (x - 5 - 8 i) (x - 5 + 8 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot 14 - 4 x - 4 \cdot 14) (x - 5 - 8 i) (x - 5 + 8 i)$

Move $14$ to the left of $x$ .

$y = (x^{2} + 14 \cdot x - 4 x - 4 \cdot 14) (x - 5 - 8 i) (x - 5 + 8 i)$

Multiply $- 4$ by $14$ .

$y = (x^{2} + 14 x - 4 x - 56) (x - 5 - 8 i) (x - 5 + 8 i)$

Subtract $4 x$ from $14 x$ .

$y = (x^{2} + 10 x - 56) (x - 5 - 8 i) (x - 5 + 8 i)$

Expand $(x^{2} + 10 x - 56) (x - 5 - 8 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 5 + x^{2} (- 8 i) + 10 x \cdot x + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot - 5 + x^{2} (- 8 i) + 10 x \cdot x + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

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

$y = (x^{2 + 1} + x^{2} \cdot - 5 + x^{2} (- 8 i) + 10 x \cdot x + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 5 + x^{2} (- 8 i) + 10 x \cdot x + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

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

$y = (x^{3} - 5 \cdot x^{2} + x^{2} (- 8 i) + 10 x \cdot x + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

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

Move $x$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 (x \cdot x) + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

Multiply $x$ by $x$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 x^{2} + 10 x \cdot - 5 + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

Multiply $- 5$ by $10$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 x^{2} - 50 x + 10 x (- 8 i) - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

Multiply $- 8$ by $10$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 x^{2} - 50 x - 80 x i - 56 x - 56 \cdot - 5 - 56 (- 8 i)) (x - 5 + 8 i)$

Multiply $- 56$ by $- 5$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 x^{2} - 50 x - 80 x i - 56 x + 280 - 56 (- 8 i)) (x - 5 + 8 i)$

Multiply $- 8$ by $- 56$ .

$y = (x^{3} - 5 x^{2} + x^{2} (- 8 i) + 10 x^{2} - 50 x - 80 x i - 56 x + 280 + 448 i) (x - 5 + 8 i)$

Simplify by adding terms.

Add $- 5 x^{2}$ and $10 x^{2}$ .

$y = (x^{3} + 5 x^{2} + x^{2} (- 8 i) - 50 x - 80 x i - 56 x + 280 + 448 i) (x - 5 + 8 i)$

Subtract $56 x$ from $- 50 x$ .

$y = (x^{3} + 5 x^{2} + x^{2} (- 8 i) - 80 x i - 106 x + 280 + 448 i) (x - 5 + 8 i)$

Expand $(x^{3} + 5 x^{2} + x^{2} (- 8 i) - 80 x i - 106 x + 280 + 448 i) (x - 5 + 8 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 5 + x^{3} (8 i) + 5 x^{2} x + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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.

$y = x^{3 + 1} + x^{3} \cdot - 5 + x^{3} (8 i) + 5 x^{2} x + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 5 + x^{3} (8 i) + 5 x^{2} x + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Move $- 5$ to the left of $x^{3}$ .

$y = x^{4} - 5 \cdot x^{3} + x^{3} (8 i) + 5 x^{2} x + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Move $8$ to the left of $x^{3}$ .

$y = x^{4} - 5 x^{3} + 8 \cdot (x^{3} i) + 5 x^{2} x + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

Move $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 (x \cdot x^{2}) + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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.

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{1 + 2} + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Add $1$ and $2$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} + 5 x^{2} \cdot - 5 + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 5$ by $5$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 5 x^{2} (8 i) + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $8$ by $5$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{2} (- 8 i) x + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

Move $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x \cdot x^{2} (- 8 i) + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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.

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{1 + 2} (- 8 i) + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Add $1$ and $2$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + x^{2} (- 8 i) \cdot - 5 + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 5$ by $- 8$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + x^{2} (40 i) + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Move $40$ to the left of $x^{2}$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 \cdot (x^{2} i) + x^{2} (- 8 i) (8 i) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $x^{2} (- 8 i) (8 i)$ .

Multiply $8$ by $- 8$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} (- 64 i) i - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} (- 64 (i i)) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} (- 64 i^{1 + 1}) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Add $1$ and $1$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} (- 64 i^{2}) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} (- 64 \cdot - 1) - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 64$ by $- 1$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + x^{2} \cdot 64 - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Move $64$ to the left of $x^{2}$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 \cdot x^{2} - 80 x i x - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

Move $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 (x \cdot x) i - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $x$ by $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i - 80 x i \cdot - 5 - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 5$ by $- 80$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 80 x i (8 i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 80 x i (8 i)$ .

Multiply $8$ by $- 80$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 640 x i i - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 640 x (i i) - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 640 x i^{1 + 1} - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Add $1$ and $1$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 640 x i^{2} - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i - 640 x \cdot - 1 - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 1$ by $- 640$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x \cdot x - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

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

Move $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 (x \cdot x) - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $x$ by $x$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} - 106 x \cdot - 5 - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 5$ by $- 106$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 106 x (8 i) + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $8$ by $- 106$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x + 280 \cdot - 5 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $280$ by $- 5$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 280 (8 i) + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $8$ by $280$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x + 448 i \cdot - 5 + 448 i (8 i)$

Multiply $- 5$ by $448$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i + 448 i (8 i)$

Multiply $448 i (8 i)$ .

Multiply $8$ by $448$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i + 3584 i i$

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i + 3584 (i i)$

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

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i + 3584 i^{1 + 1}$

Add $1$ and $1$ .

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

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i + 3584 \cdot - 1$

Multiply $3584$ by $- 1$ .

$y = x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Simplify by adding terms.

Combine the opposite terms in $x^{4} - 5 x^{3} + 8 x^{3} i + 5 x^{3} - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$ .

Add $- 5 x^{3}$ and $5 x^{3}$ .

$y = x^{4} + 8 x^{3} i + 0 - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Add $x^{4} + 8 x^{3} i$ and $0$ .

$y = x^{4} + 8 x^{3} i - 25 x^{2} + 40 x^{2} i + x^{3} (- 8 i) + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Reorder the factors in the terms $8 x^{3} i$ and $x^{3} (- 8 i)$ .

$y = x^{4} + 8 i x^{3} - 25 x^{2} + 40 x^{2} i - 8 i x^{3} + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Subtract $8 i x^{3}$ from $8 i x^{3}$ .

$y = x^{4} - 25 x^{2} + 40 x^{2} i + 0 + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Add $x^{4} - 25 x^{2} + 40 x^{2} i$ and $0$ .

$y = x^{4} - 25 x^{2} + 40 x^{2} i + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 2240 i + 448 i x - 2240 i - 3584$

Subtract $2240 i$ from $2240 i$ .

$y = x^{4} - 25 x^{2} + 40 x^{2} i + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 448 i x + 0 - 3584$

Add $x^{4} - 25 x^{2} + 40 x^{2} i + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 448 i x$ and $0$ .

$y = x^{4} - 25 x^{2} + 40 x^{2} i + 40 x^{2} i + 64 x^{2} - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Add $- 25 x^{2}$ and $64 x^{2}$ .

$y = x^{4} + 39 x^{2} + 40 x^{2} i + 40 x^{2} i - 80 x^{2} i + 400 x i + 640 x - 106 x^{2} + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Subtract $106 x^{2}$ from $39 x^{2}$ .

$y = x^{4} - 67 x^{2} + 40 x^{2} i + 40 x^{2} i - 80 x^{2} i + 400 x i + 640 x + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Add $40 x^{2} i$ and $40 x^{2} i$ .

$y = x^{4} - 67 x^{2} + 80 x^{2} i - 80 x^{2} i + 400 x i + 640 x + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Combine the opposite terms in $x^{4} - 67 x^{2} + 80 x^{2} i - 80 x^{2} i + 400 x i + 640 x + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$ .

Subtract $80 x^{2} i$ from $80 x^{2} i$ .

$y = x^{4} - 67 x^{2} + 0 + 400 x i + 640 x + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Add $x^{4} - 67 x^{2}$ and $0$ .

$y = x^{4} - 67 x^{2} + 400 x i + 640 x + 530 x - 848 x i + 280 x - 1400 + 448 i x - 3584$

Subtract $848 x i$ from $400 x i$ .

$y = x^{4} - 67 x^{2} - 448 x i + 640 x + 530 x + 280 x - 1400 + 448 i x - 3584$

Combine the opposite terms in $x^{4} - 67 x^{2} - 448 x i + 640 x + 530 x + 280 x - 1400 + 448 i x - 3584$ .

Reorder the factors in the terms $- 448 x i$ and $448 i x$ .

$y = x^{4} - 67 x^{2} - 448 i x + 640 x + 530 x + 280 x - 1400 + 448 i x - 3584$

Add $- 448 i x$ and $448 i x$ .

$y = x^{4} - 67 x^{2} + 640 x + 530 x + 280 x - 1400 + 0 - 3584$

Add $x^{4} - 67 x^{2} + 640 x + 530 x + 280 x - 1400$ and $0$ .

$y = x^{4} - 67 x^{2} + 640 x + 530 x + 280 x - 1400 - 3584$

Add $640 x$ and $530 x$ .

$y = x^{4} - 67 x^{2} + 1170 x + 280 x - 1400 - 3584$

Add $1170 x$ and $280 x$ .

$y = x^{4} - 67 x^{2} + 1450 x - 1400 - 3584$

Subtract $3584$ from $- 1400$ .

$y = x^{4} - 67 x^{2} + 1450 x - 4984$

Step 8

Find the Equation Given the Roots 4 , -14 , 5+8i

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