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

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

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

The factor is $x + 5$

Step 4

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

The factor is $x - 5 i$

Step 5

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

The factor is $x + 5 i$

Step 6

Combine all the factors into a single equation.

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

Step 7

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot 5 - 5 x - 5 \cdot 5$ .

Reorder the factors in the terms $x \cdot 5$ and $- 5 x$ .

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

Subtract $5 x$ from $5 x$ .

$y = (x \cdot x + 0 - 5 \cdot 5) (x - 5 i) (x + 5 i)$

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $5$ .

$y = (x^{2} - 25) (x - 5 i) (x + 5 i)$

Expand $(x^{2} - 25) (x - 5 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x^{2} (x - 5 i) - 25 (x - 5 i)) (x + 5 i)$

Apply the distributive property.

$y = (x^{2} x + x^{2} (- 5 i) - 25 (x - 5 i)) (x + 5 i)$

Apply the distributive property.

$y = (x^{2} x + x^{2} (- 5 i) - 25 x - 25 (- 5 i)) (x + 5 i)$

Simplify each term.

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

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

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

$y = (x^{2} x + x^{2} (- 5 i) - 25 x - 25 (- 5 i)) (x + 5 i)$

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

$y = (x^{2 + 1} + x^{2} (- 5 i) - 25 x - 25 (- 5 i)) (x + 5 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} (- 5 i) - 25 x - 25 (- 5 i)) (x + 5 i)$

Multiply $- 5$ by $- 25$ .

$y = (x^{3} + x^{2} (- 5 i) - 25 x + 125 i) (x + 5 i)$

Expand $(x^{3} + x^{2} (- 5 i) - 25 x + 125 i) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (5 i) + x^{2} (- 5 i) x + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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 $1$ .

$y = x^{3} x + x^{3} (5 i) + x^{2} (- 5 i) x + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{3 + 1} + x^{3} (5 i) + x^{2} (- 5 i) x + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (5 i) + x^{2} (- 5 i) x + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Move $5$ to the left of $x^{3}$ .

$y = x^{4} + 5 \cdot (x^{3} i) + x^{2} (- 5 i) x + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

Move $x$ .

$y = x^{4} + 5 x^{3} i + x \cdot x^{2} (- 5 i) + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

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

$y = x^{4} + 5 x^{3} i + x \cdot x^{2} (- 5 i) + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{4} + 5 x^{3} i + x^{1 + 2} (- 5 i) + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Add $1$ and $2$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 5 i) (5 i) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Multiply $x^{2} (- 5 i) (5 i)$ .

Multiply $5$ by $- 5$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 i) i - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 (i i)) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 (i i)) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 i^{1 + 1}) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Add $1$ and $1$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 i^{2}) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} (- 25 \cdot - 1) - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Multiply $- 25$ by $- 1$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + x^{2} \cdot 25 - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

Move $25$ to the left of $x^{2}$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 \cdot x^{2} - 25 x \cdot x - 25 x (5 i) + 125 i x + 125 i (5 i)$

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

Move $x$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 (x \cdot x) - 25 x (5 i) + 125 i x + 125 i (5 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 25 x (5 i) + 125 i x + 125 i (5 i)$

Multiply $5$ by $- 25$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 125 i (5 i)$

Multiply $125 i (5 i)$ .

Multiply $5$ by $125$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 i i$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 (i i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 (i i)$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 i^{2}$

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

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x + 625 \cdot - 1$

Multiply $625$ by $- 1$ .

$y = x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x - 625$

Combine the opposite terms in $x^{4} + 5 x^{3} i + x^{3} (- 5 i) + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x - 625$ .

Reorder the factors in the terms $5 x^{3} i$ and $x^{3} (- 5 i)$ .

$y = x^{4} + 5 i x^{3} - 5 i x^{3} + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x - 625$

Subtract $5 i x^{3}$ from $5 i x^{3}$ .

$y = x^{4} + 0 + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x - 625$

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

$y = x^{4} + 25 x^{2} - 25 x^{2} - 125 x i + 125 i x - 625$

Subtract $25 x^{2}$ from $25 x^{2}$ .

$y = x^{4} + 0 - 125 x i + 125 i x - 625$

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

$y = x^{4} - 125 x i + 125 i x - 625$

Reorder the factors in the terms $- 125 x i$ and $125 i x$ .

$y = x^{4} - 125 i x + 125 i x - 625$

Add $- 125 i x$ and $125 i x$ .

$y = x^{4} + 0 - 625$

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

$y = x^{4} - 625$

Step 8

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

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