Find the Equation Given the Roots -5i , 3

$- 5 i$ , $3$

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 $x = - 5 i$ was found by solving for $x$ when $x - (- 5 i) = y$ and $y = 0$ .

The factor is $x + 5 i$

Step 3

The root at $x = 5 i$ was found by solving for $x$ when $x - (5 i) = y$ and $y = 0$ .

The factor is $x - 5 i$

Step 4

The root at $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

Step 5

Combine all the factors into a single equation.

$y = (x + 5 i) (x - 5 i) (x - 3)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 5 i) (x - 5 i) (x - 3)$ .

Expand $(x + 5 i) (x - 5 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 5 i) + 5 i (x - 5 i)) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x (- 5 i) + 5 i (x - 5 i)) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x (- 5 i) + 5 i x + 5 i (- 5 i)) (x - 3)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- 5 i) + 5 i x + 5 i (- 5 i)$ .

Reorder the factors in the terms $x (- 5 i)$ and $5 i x$ .

$y = (x \cdot x - 5 i x + 5 i x + 5 i (- 5 i)) (x - 3)$

Add $- 5 i x$ and $5 i x$ .

$y = (x \cdot x + 0 + 5 i (- 5 i)) (x - 3)$

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

$y = (x \cdot x + 5 i (- 5 i)) (x - 3)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + 5 i (- 5 i)) (x - 3)$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$y = (x^{2} - 25 i i) (x - 3)$

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

$y = (x^{2} - 25 (i i)) (x - 3)$

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

$y = (x^{2} - 25 (i i)) (x - 3)$

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

$y = (x^{2} - 25 i^{1 + 1}) (x - 3)$

Add $1$ and $1$ .

$y = (x^{2} - 25 i^{2}) (x - 3)$

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

$y = (x^{2} - 25 \cdot - 1) (x - 3)$

Multiply $- 25$ by $- 1$ .

$y = (x^{2} + 25) (x - 3)$

Expand $(x^{2} + 25) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = x^{2} (x - 3) + 25 (x - 3)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 3 + 25 (x - 3)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 3 + 25 x + 25 \cdot - 3$

Simplify each term.

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

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

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

$y = x^{2} x + x^{2} \cdot - 3 + 25 x + 25 \cdot - 3$

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

$y = x^{2 + 1} + x^{2} \cdot - 3 + 25 x + 25 \cdot - 3$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 3 + 25 x + 25 \cdot - 3$

Move $- 3$ to the left of $x^{2}$ .

$y = x^{3} - 3 \cdot x^{2} + 25 x + 25 \cdot - 3$

Multiply $25$ by $- 3$ .

$y = x^{3} - 3 x^{2} + 25 x - 75$

Step 7

Find the Equation Given the Roots -5i , 3

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