Find the Equation Given the Roots -4i , 7

$- 4 i$ , $7$

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

The factor is $x + 4 i$

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 $x = 7$ was found by solving for $x$ when $x - (7) = y$ and $y = 0$ .

The factor is $x - 7$

Step 5

Combine all the factors into a single equation.

$y = (x + 4 i) (x - 4 i) (x - 7)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 4 i) (x - 4 i) (x - 7)$ .

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

Apply the distributive property.

$y = (x (x - 4 i) + 4 i (x - 4 i)) (x - 7)$

Apply the distributive property.

$y = (x \cdot x + x (- 4 i) + 4 i (x - 4 i)) (x - 7)$

Apply the distributive property.

$y = (x \cdot x + x (- 4 i) + 4 i x + 4 i (- 4 i)) (x - 7)$

Simplify terms.

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

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

$y = (x \cdot x - 4 i x + 4 i x + 4 i (- 4 i)) (x - 7)$

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

$y = (x \cdot x + 0 + 4 i (- 4 i)) (x - 7)$

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

$y = (x \cdot x + 4 i (- 4 i)) (x - 7)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + 4 i (- 4 i)) (x - 7)$

Multiply $4 i (- 4 i)$ .

Multiply $- 4$ by $4$ .

$y = (x^{2} - 16 i i) (x - 7)$

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

$y = (x^{2} - 16 (i i)) (x - 7)$

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

$y = (x^{2} - 16 (i i)) (x - 7)$

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

$y = (x^{2} - 16 i^{1 + 1}) (x - 7)$

Add $1$ and $1$ .

$y = (x^{2} - 16 i^{2}) (x - 7)$

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

$y = (x^{2} - 16 \cdot - 1) (x - 7)$

Multiply $- 16$ by $- 1$ .

$y = (x^{2} + 16) (x - 7)$

Expand $(x^{2} + 16) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$y = x^{2} (x - 7) + 16 (x - 7)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 7 + 16 (x - 7)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 7 + 16 x + 16 \cdot - 7$

Simplify each term.

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

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

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

$y = x^{2} x + x^{2} \cdot - 7 + 16 x + 16 \cdot - 7$

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

$y = x^{2 + 1} + x^{2} \cdot - 7 + 16 x + 16 \cdot - 7$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 7 + 16 x + 16 \cdot - 7$

Move $- 7$ to the left of $x^{2}$ .

$y = x^{3} - 7 \cdot x^{2} + 16 x + 16 \cdot - 7$

Multiply $16$ by $- 7$ .

$y = x^{3} - 7 x^{2} + 16 x - 112$

Step 7

Find the Equation Given the Roots -4i , 7

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