Find the Equation Given the Roots 5 , 9i , -9i

$5$ , $9 i$ , $- 9 i$

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

The factor is $x - 5$

Step 3

The root at $x = 9 i$ was found by solving for $x$ when $x - (9 i) = y$ and $y = 0$ .

The factor is $x - 9 i$

Step 4

The root at $x = - 9 i$ was found by solving for $x$ when $x - (- 9 i) = y$ and $y = 0$ .

The factor is $x + 9 i$

Step 5

Combine all the factors into a single equation.

$y = (x - 5) (x - 9 i) (x + 9 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 5) (x - 9 i) (x + 9 i)$ .

Expand $(x - 5) (x - 9 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 9 i) - 5 (x - 9 i)) (x + 9 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 9 i) - 5 (x - 9 i)) (x + 9 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 9 i) - 5 x - 5 (- 9 i)) (x + 9 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x (- 9 i) - 5 x - 5 (- 9 i)) (x + 9 i)$

Multiply $- 9$ by $- 5$ .

$y = (x^{2} + x (- 9 i) - 5 x + 45 i) (x + 9 i)$

Expand $(x^{2} + x (- 9 i) - 5 x + 45 i) (x + 9 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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 $1$ .

$y = x^{2} x + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

$y = x^{2 + 1} + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Move $9$ to the left of $x^{2}$ .

$y = x^{3} + 9 \cdot (x^{2} i) + x (- 9 i) x + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

Move $x$ .

$y = x^{3} + 9 x^{2} i + x \cdot x (- 9 i) + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 9 i) (9 i) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Multiply $x (- 9 i) (9 i)$ .

Multiply $9$ by $- 9$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i) i - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 (i i)) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 (i i)) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i^{1 + 1}) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Add $1$ and $1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i^{2}) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 \cdot - 1) - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Multiply $- 81$ by $- 1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x \cdot 81 - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

Move $81$ to the left of $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 \cdot x - 5 x \cdot x - 5 x (9 i) + 45 i x + 45 i (9 i)$

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

Move $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 (x \cdot x) - 5 x (9 i) + 45 i x + 45 i (9 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 5 x (9 i) + 45 i x + 45 i (9 i)$

Multiply $9$ by $- 5$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 45 i (9 i)$

Multiply $45 i (9 i)$ .

Multiply $9$ by $45$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 i i$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 (i i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 (i i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 i^{2}$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x + 405 \cdot - 1$

Multiply $405$ by $- 1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x - 405$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 5 x^{2} - 45 x i + 45 i x - 405$ .

Reorder the factors in the terms $9 x^{2} i$ and $x^{2} (- 9 i)$ .

$y = x^{3} + 9 i x^{2} - 9 i x^{2} + 81 x - 5 x^{2} - 45 x i + 45 i x - 405$

Subtract $9 i x^{2}$ from $9 i x^{2}$ .

$y = x^{3} + 0 + 81 x - 5 x^{2} - 45 x i + 45 i x - 405$

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

$y = x^{3} + 81 x - 5 x^{2} - 45 x i + 45 i x - 405$

Reorder the factors in the terms $- 45 x i$ and $45 i x$ .

$y = x^{3} + 81 x - 5 x^{2} - 45 i x + 45 i x - 405$

Add $- 45 i x$ and $45 i x$ .

$y = x^{3} + 81 x - 5 x^{2} + 0 - 405$

Add $x^{3} + 81 x - 5 x^{2}$ and $0$ .

$y = x^{3} + 81 x - 5 x^{2} - 405$

Move $81 x$ .

$y = x^{3} - 5 x^{2} + 81 x - 405$

Step 7

Find the Equation Given the Roots 5 , 9i , -9i

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