Find the Equation Given the Roots -1 , 4i

$- 1$ , $4 i$

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 $x = - 1$ was found by solving for $x$ when $x - (- 1) = y$ and $y = 0$ .

The factor is $x + 1$

Step 3

The root at $x = 4 i$ 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.

$y = (x + 1) (x - 4 i) (x + 4 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 1) (x - 4 i) (x + 4 i)$ .

Expand $(x + 1) (x - 4 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 4 i) + 1 (x - 4 i)) (x + 4 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 4 i) + 1 (x - 4 i)) (x + 4 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 4 i) + 1 x + 1 (- 4 i)) (x + 4 i)$

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 4 i$ by $1$ .

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

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

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

Simplify terms.

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

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

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

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

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

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

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

Multiply $4$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 16$ by $- 1$ .

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

Move $16$ to the left of $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4 i (4 i)$ .

Multiply $4$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 16$ by $- 1$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

Move $16 x$ .

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

Step 7

Find the Equation Given the Roots -1 , 4i

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