Find the Equation Given the Roots square root of 2 , 2i

$\sqrt{2}$ , $2 i$

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

The factor is $x - \sqrt{2}$

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.

$y = (x - \sqrt{2}) (x - 2 i) (x + 2 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - \sqrt{2}) (x - 2 i) (x + 2 i)$ .

Expand $(x - \sqrt{2}) (x - 2 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 2 i) - \sqrt{2} (x - 2 i)) (x + 2 i)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $2 \sqrt{2} i (2 i)$ .

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $4$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

Reorder the factors in the terms $- 2 \sqrt{2} x i$ and $2 \sqrt{2} i x$ .

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

Add $- 2 i x \sqrt{2}$ and $2 i x \sqrt{2}$ .

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

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

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

Move $4 x$ .

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

Step 7

Find the Equation Given the Roots square root of 2 , 2i

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