Find the Equation Given the Roots 10 , -10 , 10i , -10i

$10$ , $- 10$ , $10 i$ , $- 10 i$

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 $x = 10$ was found by solving for $x$ when $x - (10) = y$ and $y = 0$ .

The factor is $x - 10$

Step 3

The root at $x = - 10$ was found by solving for $x$ when $x - (- 10) = y$ and $y = 0$ .

The factor is $x + 10$

Step 4

The root at $x = 10 i$ was found by solving for $x$ when $x - (10 i) = y$ and $y = 0$ .

The factor is $x - 10 i$

Step 5

The root at $x = - 10 i$ was found by solving for $x$ when $x - (- 10 i) = y$ and $y = 0$ .

The factor is $x + 10 i$

Step 6

Combine all the factors into a single equation.

$y = (x - 10) (x + 10) (x - 10 i) (x + 10 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x - 10) (x + 10) (x - 10 i) (x + 10 i)$ .

Expand $(x - 10) (x + 10)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 10) - 10 (x + 10)) (x - 10 i) (x + 10 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 10 - 10 (x + 10)) (x - 10 i) (x + 10 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 10 - 10 x - 10 \cdot 10) (x - 10 i) (x + 10 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot 10 - 10 x - 10 \cdot 10$ .

Reorder the factors in the terms $x \cdot 10$ and $- 10 x$ .

$y = (x \cdot x + 10 x - 10 x - 10 \cdot 10) (x - 10 i) (x + 10 i)$

Subtract $10 x$ from $10 x$ .

$y = (x \cdot x + 0 - 10 \cdot 10) (x - 10 i) (x + 10 i)$

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

$y = (x \cdot x - 10 \cdot 10) (x - 10 i) (x + 10 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} - 10 \cdot 10) (x - 10 i) (x + 10 i)$

Multiply $- 10$ by $10$ .

$y = (x^{2} - 100) (x - 10 i) (x + 10 i)$

Expand $(x^{2} - 100) (x - 10 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x^{2} (x - 10 i) - 100 (x - 10 i)) (x + 10 i)$

Apply the distributive property.

$y = (x^{2} x + x^{2} (- 10 i) - 100 (x - 10 i)) (x + 10 i)$

Apply the distributive property.

$y = (x^{2} x + x^{2} (- 10 i) - 100 x - 100 (- 10 i)) (x + 10 i)$

Simplify each term.

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

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

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

$y = (x^{2} x + x^{2} (- 10 i) - 100 x - 100 (- 10 i)) (x + 10 i)$

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

$y = (x^{2 + 1} + x^{2} (- 10 i) - 100 x - 100 (- 10 i)) (x + 10 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} (- 10 i) - 100 x - 100 (- 10 i)) (x + 10 i)$

Multiply $- 10$ by $- 100$ .

$y = (x^{3} + x^{2} (- 10 i) - 100 x + 1000 i) (x + 10 i)$

Expand $(x^{3} + x^{2} (- 10 i) - 100 x + 1000 i) (x + 10 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (10 i) + x^{2} (- 10 i) x + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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 $1$ .

$y = x^{3} x + x^{3} (10 i) + x^{2} (- 10 i) x + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{3 + 1} + x^{3} (10 i) + x^{2} (- 10 i) x + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (10 i) + x^{2} (- 10 i) x + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Move $10$ to the left of $x^{3}$ .

$y = x^{4} + 10 \cdot (x^{3} i) + x^{2} (- 10 i) x + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

Move $x$ .

$y = x^{4} + 10 x^{3} i + x \cdot x^{2} (- 10 i) + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

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

$y = x^{4} + 10 x^{3} i + x \cdot x^{2} (- 10 i) + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{4} + 10 x^{3} i + x^{1 + 2} (- 10 i) + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Add $1$ and $2$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 10 i) (10 i) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Multiply $x^{2} (- 10 i) (10 i)$ .

Multiply $10$ by $- 10$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 i) i - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 (i i)) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 (i i)) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 i^{1 + 1}) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Add $1$ and $1$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 i^{2}) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} (- 100 \cdot - 1) - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Multiply $- 100$ by $- 1$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + x^{2} \cdot 100 - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Move $100$ to the left of $x^{2}$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 \cdot x^{2} - 100 x \cdot x - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

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

Move $x$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 (x \cdot x) - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 100 x (10 i) + 1000 i x + 1000 i (10 i)$

Multiply $10$ by $- 100$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 1000 i (10 i)$

Multiply $1000 i (10 i)$ .

Multiply $10$ by $1000$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 i i$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 (i i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 (i i)$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 i^{2}$

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

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x + 10000 \cdot - 1$

Multiply $10000$ by $- 1$ .

$y = x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x - 10000$

Combine the opposite terms in $x^{4} + 10 x^{3} i + x^{3} (- 10 i) + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x - 10000$ .

Reorder the factors in the terms $10 x^{3} i$ and $x^{3} (- 10 i)$ .

$y = x^{4} + 10 i x^{3} - 10 i x^{3} + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x - 10000$

Subtract $10 i x^{3}$ from $10 i x^{3}$ .

$y = x^{4} + 0 + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x - 10000$

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

$y = x^{4} + 100 x^{2} - 100 x^{2} - 1000 x i + 1000 i x - 10000$

Subtract $100 x^{2}$ from $100 x^{2}$ .

$y = x^{4} + 0 - 1000 x i + 1000 i x - 10000$

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

$y = x^{4} - 1000 x i + 1000 i x - 10000$

Reorder the factors in the terms $- 1000 x i$ and $1000 i x$ .

$y = x^{4} - 1000 i x + 1000 i x - 10000$

Add $- 1000 i x$ and $1000 i x$ .

$y = x^{4} + 0 - 10000$

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

$y = x^{4} - 10000$

Step 8

Find the Equation Given the Roots 10 , -10 , 10i , -10i

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