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

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

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

The factor is $x + 2$

Step 3

The root at $x = 1 + 2 i$ was found by solving for $x$ when $x - (1 + 2 i) = y$ and $y = 0$ .

The factor is $x - 1 - 2 i$

Step 4

The root at $x = 1 - 2 i$ was found by solving for $x$ when $x - (1 - 2 i) = y$ and $y = 0$ .

The factor is $x - 1 + 2 i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Multiply $- 2$ by $2$ .

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

Add $- x$ and $2 x$ .

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

Expand $(x^{2} + x (- 2 i) + x - 2 - 4 i) (x - 1 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $1$ .

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 2$ .

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

Move $2$ to the left of $x$ .

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

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Move $4$ to the left of $x$ .

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

Multiply $x$ by $x$ .

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

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

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

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

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

Move $2$ to the left of $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $- 1$ by $- 4$ .

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

Multiply $- 4 i (2 i)$ .

Multiply $2$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 8$ by $- 1$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

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

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

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

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

Add $- 4 i$ and $4 i$ .

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

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

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

Add $2 x i$ and $2 x i$ .

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

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

Reorder the factors in the terms $4 x i$ and $- 4 i x$ .

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

Subtract $4 i x$ from $4 i x$ .

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

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

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

Subtract $x$ from $4 x$ .

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

Subtract $2 x$ from $3 x$ .

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

Add $2$ and $8$ .

$y = x^{3} + x + 10$

Step 7

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

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