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

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

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

The factor is $x - 9$

Step 3

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

The factor is $x + 9$

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

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 6

Combine all the factors into a single equation.

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

Step 7

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot 9 - 9 x - 9 \cdot 9$ .

Reorder the factors in the terms $x \cdot 9$ and $- 9 x$ .

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

Subtract $9 x$ from $9 x$ .

$y = (x \cdot x + 0 - 9 \cdot 9) (x - 9 i) (x + 9 i)$

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 9$ by $9$ .

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

Expand $(x^{2} - 81) (x - 9 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 9$ by $- 81$ .

$y = (x^{3} + x^{2} (- 9 i) - 81 x + 729 i) (x + 9 i)$

Expand $(x^{3} + x^{2} (- 9 i) - 81 x + 729 i) (x + 9 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (9 i) + x^{2} (- 9 i) x + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$y = x^{3} x + x^{3} (9 i) + x^{2} (- 9 i) x + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{3 + 1} + x^{3} (9 i) + x^{2} (- 9 i) x + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (9 i) + x^{2} (- 9 i) x + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Move $9$ to the left of $x^{3}$ .

$y = x^{4} + 9 \cdot (x^{3} i) + x^{2} (- 9 i) x + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

Move $x$ .

$y = x^{4} + 9 x^{3} i + x \cdot x^{2} (- 9 i) + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

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

$y = x^{4} + 9 x^{3} i + x \cdot x^{2} (- 9 i) + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{4} + 9 x^{3} i + x^{1 + 2} (- 9 i) + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Add $1$ and $2$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 9 i) (9 i) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Multiply $x^{2} (- 9 i) (9 i)$ .

Multiply $9$ by $- 9$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 i) i - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 (i i)) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 (i i)) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 i^{1 + 1}) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Add $1$ and $1$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 i^{2}) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} (- 81 \cdot - 1) - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Multiply $- 81$ by $- 1$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + x^{2} \cdot 81 - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

Move $81$ to the left of $x^{2}$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 \cdot x^{2} - 81 x \cdot x - 81 x (9 i) + 729 i x + 729 i (9 i)$

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

Move $x$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 (x \cdot x) - 81 x (9 i) + 729 i x + 729 i (9 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 81 x (9 i) + 729 i x + 729 i (9 i)$

Multiply $9$ by $- 81$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 729 i (9 i)$

Multiply $729 i (9 i)$ .

Multiply $9$ by $729$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 i i$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 (i i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 (i i)$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 i^{2}$

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

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x + 6561 \cdot - 1$

Multiply $6561$ by $- 1$ .

$y = x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x - 6561$

Combine the opposite terms in $x^{4} + 9 x^{3} i + x^{3} (- 9 i) + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x - 6561$ .

Reorder the factors in the terms $9 x^{3} i$ and $x^{3} (- 9 i)$ .

$y = x^{4} + 9 i x^{3} - 9 i x^{3} + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x - 6561$

Subtract $9 i x^{3}$ from $9 i x^{3}$ .

$y = x^{4} + 0 + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x - 6561$

Add $x^{4}$ and $0$ .

$y = x^{4} + 81 x^{2} - 81 x^{2} - 729 x i + 729 i x - 6561$

Subtract $81 x^{2}$ from $81 x^{2}$ .

$y = x^{4} + 0 - 729 x i + 729 i x - 6561$

Add $x^{4}$ and $0$ .

$y = x^{4} - 729 x i + 729 i x - 6561$

Reorder the factors in the terms $- 729 x i$ and $729 i x$ .

$y = x^{4} - 729 i x + 729 i x - 6561$

Add $- 729 i x$ and $729 i x$ .

$y = x^{4} + 0 - 6561$

Add $x^{4}$ and $0$ .

$y = x^{4} - 6561$

Step 8

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

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