Find the Equation Given the Roots 3 , 9i , -9i

$3$ , $9 i$ , $- 9 i$

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

The factor is $x - 3$

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.

$y = (x - 3) (x - 9 i) (x + 9 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3) (x - 9 i) (x + 9 i)$ .

Expand $(x - 3) (x - 9 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 9 i) - 3 (x - 9 i)) (x + 9 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 9 i) - 3 (x - 9 i)) (x + 9 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 9 i) - 3 x - 3 (- 9 i)) (x + 9 i)$

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 9$ by $- 3$ .

$y = (x^{2} + x (- 9 i) - 3 x + 27 i) (x + 9 i)$

Expand $(x^{2} + x (- 9 i) - 3 x + 27 i) (x + 9 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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 $1$ .

$y = x^{2} x + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{2 + 1} + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} (9 i) + x (- 9 i) x + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 \cdot (x^{2} i) + x (- 9 i) x + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

Move $x$ .

$y = x^{3} + 9 x^{2} i + x \cdot x (- 9 i) + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 9 i) (9 i) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

Multiply $9$ by $- 9$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i) i - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 (i i)) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 (i i)) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i^{1 + 1}) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

Add $1$ and $1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 i^{2}) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x (- 81 \cdot - 1) - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

Multiply $- 81$ by $- 1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + x \cdot 81 - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 \cdot x - 3 x \cdot x - 3 x (9 i) + 27 i x + 27 i (9 i)$

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

Move $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 (x \cdot x) - 3 x (9 i) + 27 i x + 27 i (9 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 3 x (9 i) + 27 i x + 27 i (9 i)$

Multiply $9$ by $- 3$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 27 i (9 i)$

Multiply $27 i (9 i)$ .

Multiply $9$ by $27$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 i i$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 (i i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 (i i)$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 i^{2}$

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

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x + 243 \cdot - 1$

Multiply $243$ by $- 1$ .

$y = x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x - 243$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 9 x^{2} i + x^{2} (- 9 i) + 81 x - 3 x^{2} - 27 x i + 27 i x - 243$ .

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

$y = x^{3} + 9 i x^{2} - 9 i x^{2} + 81 x - 3 x^{2} - 27 x i + 27 i x - 243$

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

$y = x^{3} + 0 + 81 x - 3 x^{2} - 27 x i + 27 i x - 243$

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

$y = x^{3} + 81 x - 3 x^{2} - 27 x i + 27 i x - 243$

Reorder the factors in the terms $- 27 x i$ and $27 i x$ .

$y = x^{3} + 81 x - 3 x^{2} - 27 i x + 27 i x - 243$

Add $- 27 i x$ and $27 i x$ .

$y = x^{3} + 81 x - 3 x^{2} + 0 - 243$

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

$y = x^{3} + 81 x - 3 x^{2} - 243$

Move $81 x$ .

$y = x^{3} - 3 x^{2} + 81 x - 243$

Step 7

Find the Equation Given the Roots 3 , 9i , -9i

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