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

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

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

The factor is $x - 2 i$

Step 4

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

The factor is $x + 2 i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

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

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 2 i$ by $1$ .

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

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

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

Simplify terms.

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

Reorder the factors in the terms $x (2 i)$ and $- 2 i x$ .

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

Subtract $2 i x$ from $2 i x$ .

$y = x^{3} x + x^{3} (2 i) + x^{2} (- 2 i) x + x^{2} (- 2 i) (2 i) + 2 x^{2} x + 2 x^{2} (2 i) - 4 x i x - 4 x i (2 i) + x \cdot x + 0 - 2 i (2 i)$

Add $x^{3} x + x^{3} (2 i) + x^{2} (- 2 i) x + x^{2} (- 2 i) (2 i) + 2 x^{2} x + 2 x^{2} (2 i) - 4 x i x - 4 x i (2 i) + x \cdot x$ and $0$ .

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

Simplify each term.

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

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

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

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

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

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

Add $3$ and $1$ .

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

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

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

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

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $2$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4 x i (2 i)$ .

Multiply $2$ by $- 4$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x i i + x \cdot x - 2 i (2 i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x (i i) + x \cdot x - 2 i (2 i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x (i i) + x \cdot x - 2 i (2 i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x i^{1 + 1} + x \cdot x - 2 i (2 i)$

Add $1$ and $1$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x i^{2} + x \cdot x - 2 i (2 i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i - 8 x \cdot - 1 + x \cdot x - 2 i (2 i)$

Multiply $- 1$ by $- 8$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x \cdot x - 2 i (2 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 2 i (2 i)$

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 i i$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 (i i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 (i i)$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 i^{2}$

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

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$y = x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} + 4$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + 2 x^{3} i + x^{3} (- 2 i) + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} + 4$ .

Reorder the factors in the terms $2 x^{3} i$ and $x^{3} (- 2 i)$ .

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

Subtract $2 i x^{3}$ from $2 i x^{3}$ .

$y = x^{4} + 0 + 4 x^{2} + 2 x^{3} + 4 x^{2} i - 4 x^{2} i + 8 x + x^{2} + 4$

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

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

Subtract $4 x^{2} i$ from $4 x^{2} i$ .

$y = x^{4} + 4 x^{2} + 2 x^{3} + 0 + 8 x + x^{2} + 4$

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

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

Add $4 x^{2}$ and $x^{2}$ .

$y = x^{4} + 5 x^{2} + 2 x^{3} + 8 x + 4$

Move $5 x^{2}$ .

$y = x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 4$

Step 7

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

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