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

$- 4$ , $- 4 i$

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

The factor is $x + 4$

Step 3

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 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

Combine all the factors into a single equation.

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

Step 6

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

Expand $(x + 4) (x + 4 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $4$ by $4$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

Multiply $- 4$ by $4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 16$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4$ by $4$ .

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

Multiply $16 i (- 4 i)$ .

Multiply $- 4$ by $16$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 64$ by $- 1$ .

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

Simplify by adding terms.

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

Reorder the factors in the terms $x^{2} (- 4 i)$ and $4 x^{2} i$ .

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

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

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

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

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

Reorder the factors in the terms $- 16 x i$ and $16 i x$ .

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

Add $- 16 i x$ and $16 i x$ .

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

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

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

Move $16 x$ .

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

Step 7

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

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