Find the Equation Given the Roots -1 , 5 , -4i

$- 1$ , $5$ , $- 4 i$

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

The factor is $x + 1$

Step 3

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

The factor is $x - 5$

Step 4

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

The factor is $x + 4 i$

Step 5

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

The factor is $x - 4 i$

Step 6

Combine all the factors into a single equation.

$y = (x + 1) (x - 5) (x + 4 i) (x - 4 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x + 1) (x - 5) (x + 4 i) (x - 4 i)$ .

Expand $(x + 1) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 5) + 1 (x - 5)) (x + 4 i) (x - 4 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 5 + 1 (x - 5)) (x + 4 i) (x - 4 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 5 + 1 x + 1 \cdot - 5) (x + 4 i) (x - 4 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 5$ to the left of $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 5$ by $1$ .

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

Add $- 5 x$ and $x$ .

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

Expand $(x^{2} - 4 x - 5) (x + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

Move $4$ to the left of $x^{2}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $4$ by $- 4$ .

$y = (x^{3} + 4 x^{2} i - 4 x^{2} - 16 x i - 5 x - 5 (4 i)) (x - 4 i)$

Multiply $4$ by $- 5$ .

$y = (x^{3} + 4 x^{2} i - 4 x^{2} - 16 x i - 5 x - 20 i) (x - 4 i)$

Expand $(x^{3} + 4 x^{2} i - 4 x^{2} - 16 x i - 5 x - 20 i) (x - 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (- 4 i) + 4 x^{2} i x + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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 $1$ .

$y = x^{3} x + x^{3} (- 4 i) + 4 x^{2} i x + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{3 + 1} + x^{3} (- 4 i) + 4 x^{2} i x + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{2} i x + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 (x \cdot x^{2}) i + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

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

$y = x^{4} + x^{3} (- 4 i) + 4 (x \cdot x^{2}) i + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{1 + 2} i + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Add $1$ and $2$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 4 x^{2} i (- 4 i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $4 x^{2} i (- 4 i)$ .

Multiply $- 4$ by $4$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} i i - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} (i i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} (i i) - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} i^{1 + 1} - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} i^{2} - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i - 16 x^{2} \cdot - 1 - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $- 1$ by $- 16$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{2} x - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 (x \cdot x^{2}) - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 (x \cdot x^{2}) - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{1 + 2} - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Add $1$ and $2$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} - 4 x^{2} (- 4 i) - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $- 4$ by $- 4$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x i x - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 (x \cdot x) i - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $x$ by $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 16 x i (- 4 i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $- 16 x i (- 4 i)$ .

Multiply $- 4$ by $- 16$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x i i - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x (i i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x (i i) - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x i^{1 + 1} - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x i^{2} - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i + 64 x \cdot - 1 - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $- 1$ by $64$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x \cdot x - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 (x \cdot x) - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $x$ by $x$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} - 5 x (- 4 i) - 20 i x - 20 i (- 4 i)$

Multiply $- 4$ by $- 5$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 20 i (- 4 i)$

Multiply $- 20 i (- 4 i)$ .

Multiply $- 4$ by $- 20$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 i i$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 (i i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 (i i)$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 i^{2}$

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

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x + 80 \cdot - 1$

Multiply $80$ by $- 1$ .

$y = x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + x^{3} (- 4 i) + 4 x^{3} i + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$ .

Reorder the factors in the terms $x^{3} (- 4 i)$ and $4 x^{3} i$ .

$y = x^{4} - 4 i x^{3} + 4 i x^{3} + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

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

$y = x^{4} + 0 + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

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

$y = x^{4} + 16 x^{2} - 4 x^{3} + 16 x^{2} i - 16 x^{2} i - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

Subtract $16 x^{2} i$ from $16 x^{2} i$ .

$y = x^{4} + 16 x^{2} - 4 x^{3} + 0 - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

Add $x^{4} + 16 x^{2} - 4 x^{3}$ and $0$ .

$y = x^{4} + 16 x^{2} - 4 x^{3} - 64 x - 5 x^{2} + 20 x i - 20 i x - 80$

Reorder the factors in the terms $20 x i$ and $- 20 i x$ .

$y = x^{4} + 16 x^{2} - 4 x^{3} - 64 x - 5 x^{2} + 20 i x - 20 i x - 80$

Subtract $20 i x$ from $20 i x$ .

$y = x^{4} + 16 x^{2} - 4 x^{3} - 64 x - 5 x^{2} + 0 - 80$

Add $x^{4} + 16 x^{2} - 4 x^{3} - 64 x - 5 x^{2}$ and $0$ .

$y = x^{4} + 16 x^{2} - 4 x^{3} - 64 x - 5 x^{2} - 80$

Subtract $5 x^{2}$ from $16 x^{2}$ .

$y = x^{4} + 11 x^{2} - 4 x^{3} - 64 x - 80$

Move $11 x^{2}$ .

$y = x^{4} - 4 x^{3} + 11 x^{2} - 64 x - 80$

Step 8

Find the Equation Given the Roots -1 , 5 , -4i

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