Find the Equation Given the Roots -2i , 4

$- 2 i$ , $4$

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

The factor is $x + 2 i$

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

The factor is $x - 4$

Step 5

Combine all the factors into a single equation.

$y = (x + 2 i) (x - 2 i) (x - 4)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 2 i) (x - 2 i) (x - 4)$ .

Expand $(x + 2 i) (x - 2 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 2 i) + 2 i (x - 2 i)) (x - 4)$

Apply the distributive property.

$y = (x \cdot x + x (- 2 i) + 2 i (x - 2 i)) (x - 4)$

Apply the distributive property.

$y = (x \cdot x + x (- 2 i) + 2 i x + 2 i (- 2 i)) (x - 4)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- 2 i) + 2 i x + 2 i (- 2 i)$ .

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

$y = (x \cdot x - 2 i x + 2 i x + 2 i (- 2 i)) (x - 4)$

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

$y = (x \cdot x + 0 + 2 i (- 2 i)) (x - 4)$

Add $x \cdot x$ and $0$ .

$y = (x \cdot x + 2 i (- 2 i)) (x - 4)$

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Expand $(x^{2} + 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $4$ by $- 4$ .

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

Step 7

Find the Equation Given the Roots -2i , 4

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