Find the Equation Given the Roots 4 , -8i

$4$ , $- 8 i$

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 $x = 4$ was found by solving for $x$ when $x - (4) = y$ and $y = 0$ .

The factor is $x - 4$

Step 3

The root at $x = - 8 i$ was found by solving for $x$ when $x - (- 8 i) = y$ and $y = 0$ .

The factor is $x + 8 i$

Step 4

The root at $x = 8 i$ was found by solving for $x$ when $x - (8 i) = y$ and $y = 0$ .

The factor is $x - 8 i$

Step 5

Combine all the factors into a single equation.

$y = (x - 4) (x + 8 i) (x - 8 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 4) (x + 8 i) (x - 8 i)$ .

Expand $(x - 4) (x + 8 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 8 i) - 4 (x + 8 i)) (x - 8 i)$

Apply the distributive property.

$y = (x \cdot x + x (8 i) - 4 (x + 8 i)) (x - 8 i)$

Apply the distributive property.

$y = (x \cdot x + x (8 i) - 4 x - 4 (8 i)) (x - 8 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x (8 i) - 4 x - 4 (8 i)) (x - 8 i)$

Move $8$ to the left of $x$ .

$y = (x^{2} + 8 \cdot (x i) - 4 x - 4 (8 i)) (x - 8 i)$

Multiply $8$ by $- 4$ .

$y = (x^{2} + 8 x i - 4 x - 32 i) (x - 8 i)$

Expand $(x^{2} + 8 x i - 4 x - 32 i) (x - 8 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (- 8 i) + 8 x i x + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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 $1$ .

$y = x^{2} x + x^{2} (- 8 i) + 8 x i x + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

$y = x^{2 + 1} + x^{2} (- 8 i) + 8 x i x + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x i x + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

Move $x$ .

$y = x^{3} + x^{2} (- 8 i) + 8 (x \cdot x) i + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Multiply $x$ by $x$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 8 x i (- 8 i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Multiply $8 x i (- 8 i)$ .

Multiply $- 8$ by $8$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x i i - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x (i i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x (i i) - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x i^{1 + 1} - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Add $1$ and $1$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x i^{2} - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i - 64 x \cdot - 1 - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Multiply $- 1$ by $- 64$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x \cdot x - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

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

Move $x$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 (x \cdot x) - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Multiply $x$ by $x$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} - 4 x (- 8 i) - 32 i x - 32 i (- 8 i)$

Multiply $- 8$ by $- 4$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x - 32 i (- 8 i)$

Multiply $- 32 i (- 8 i)$ .

Multiply $- 8$ by $- 32$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 i i$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 (i i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 (i i)$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 i^{2}$

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

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x + 256 \cdot - 1$

Multiply $256$ by $- 1$ .

$y = x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x - 256$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + x^{2} (- 8 i) + 8 x^{2} i + 64 x - 4 x^{2} + 32 x i - 32 i x - 256$ .

Reorder the factors in the terms $x^{2} (- 8 i)$ and $8 x^{2} i$ .

$y = x^{3} - 8 i x^{2} + 8 i x^{2} + 64 x - 4 x^{2} + 32 x i - 32 i x - 256$

Add $- 8 i x^{2}$ and $8 i x^{2}$ .

$y = x^{3} + 0 + 64 x - 4 x^{2} + 32 x i - 32 i x - 256$

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

$y = x^{3} + 64 x - 4 x^{2} + 32 x i - 32 i x - 256$

Reorder the factors in the terms $32 x i$ and $- 32 i x$ .

$y = x^{3} + 64 x - 4 x^{2} + 32 i x - 32 i x - 256$

Subtract $32 i x$ from $32 i x$ .

$y = x^{3} + 64 x - 4 x^{2} + 0 - 256$

Add $x^{3} + 64 x - 4 x^{2}$ and $0$ .

$y = x^{3} + 64 x - 4 x^{2} - 256$

Move $64 x$ .

$y = x^{3} - 4 x^{2} + 64 x - 256$

Step 7

Find the Equation Given the Roots 4 , -8i

Related Questions

Support LCMGCF.com by disabling your adblocker.