Find the Equation Given the Roots 5 , 10i , -10i

$5$ , $10 i$ , $- 10 i$

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

The factor is $x - 5$

Step 3

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

The factor is $x - 10 i$

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

Combine all the factors into a single equation.

$y = (x - 5) (x - 10 i) (x + 10 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 5) (x - 10 i) (x + 10 i)$ .

Expand $(x - 5) (x - 10 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 10 i) - 5 (x - 10 i)) (x + 10 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 10 i) - 5 (x - 10 i)) (x + 10 i)$

Apply the distributive property.

$y = (x \cdot x + x (- 10 i) - 5 x - 5 (- 10 i)) (x + 10 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x (- 10 i) - 5 x - 5 (- 10 i)) (x + 10 i)$

Multiply $- 10$ by $- 5$ .

$y = (x^{2} + x (- 10 i) - 5 x + 50 i) (x + 10 i)$

Expand $(x^{2} + x (- 10 i) - 5 x + 50 i) (x + 10 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (10 i) + x (- 10 i) x + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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 $1$ .

$y = x^{2} x + x^{2} (10 i) + x (- 10 i) x + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

$y = x^{2 + 1} + x^{2} (10 i) + x (- 10 i) x + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} (10 i) + x (- 10 i) x + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Move $10$ to the left of $x^{2}$ .

$y = x^{3} + 10 \cdot (x^{2} i) + x (- 10 i) x + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

Move $x$ .

$y = x^{3} + 10 x^{2} i + x \cdot x (- 10 i) + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 10 i) (10 i) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Multiply $x (- 10 i) (10 i)$ .

Multiply $10$ by $- 10$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 i) i - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 (i i)) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 (i i)) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 i^{1 + 1}) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Add $1$ and $1$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 i^{2}) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x (- 100 \cdot - 1) - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Multiply $- 100$ by $- 1$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + x \cdot 100 - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

Move $100$ to the left of $x$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 \cdot x - 5 x \cdot x - 5 x (10 i) + 50 i x + 50 i (10 i)$

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

Move $x$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 (x \cdot x) - 5 x (10 i) + 50 i x + 50 i (10 i)$

Multiply $x$ by $x$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 5 x (10 i) + 50 i x + 50 i (10 i)$

Multiply $10$ by $- 5$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 50 i (10 i)$

Multiply $50 i (10 i)$ .

Multiply $10$ by $50$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 i i$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 (i i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 (i i)$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 i^{2}$

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

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x + 500 \cdot - 1$

Multiply $500$ by $- 1$ .

$y = x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x - 500$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 10 x^{2} i + x^{2} (- 10 i) + 100 x - 5 x^{2} - 50 x i + 50 i x - 500$ .

Reorder the factors in the terms $10 x^{2} i$ and $x^{2} (- 10 i)$ .

$y = x^{3} + 10 i x^{2} - 10 i x^{2} + 100 x - 5 x^{2} - 50 x i + 50 i x - 500$

Subtract $10 i x^{2}$ from $10 i x^{2}$ .

$y = x^{3} + 0 + 100 x - 5 x^{2} - 50 x i + 50 i x - 500$

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

$y = x^{3} + 100 x - 5 x^{2} - 50 x i + 50 i x - 500$

Reorder the factors in the terms $- 50 x i$ and $50 i x$ .

$y = x^{3} + 100 x - 5 x^{2} - 50 i x + 50 i x - 500$

Add $- 50 i x$ and $50 i x$ .

$y = x^{3} + 100 x - 5 x^{2} + 0 - 500$

Add $x^{3} + 100 x - 5 x^{2}$ and $0$ .

$y = x^{3} + 100 x - 5 x^{2} - 500$

Move $100 x$ .

$y = x^{3} - 5 x^{2} + 100 x - 500$

Step 7

Find the Equation Given the Roots 5 , 10i , -10i

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