Find the Equation Given the Roots 4+5i , 4-5i , -1

$4 + 5 i$ , $4 - 5 i$ , $- 1$

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 $x = 4 + 5 i$ was found by solving for $x$ when $x - (4 + 5 i) = y$ and $y = 0$ .

The factor is $x - 4 - 5 i$

Step 3

The root at $x = 4 - 5 i$ was found by solving for $x$ when $x - (4 - 5 i) = y$ and $y = 0$ .

The factor is $x - 4 + 5 i$

Step 4

The root at $x = - 1$ was found by solving for $x$ when $x - (- 1) = y$ and $y = 0$ .

The factor is $x + 1$

Step 5

Combine all the factors into a single equation.

$y = (x - 4 - 5 i) (x - 4 + 5 i) (x + 1)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 4 - 5 i) (x - 4 + 5 i) (x + 1)$ .

Expand $(x - 4 - 5 i) (x - 4 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 4 + x (5 i) - 4 x - 4 \cdot - 4 - 4 (5 i) - 5 i x - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 4 + x (5 i) - 4 x - 4 \cdot - 4 - 4 (5 i) - 5 i x - 5 i \cdot - 4 - 5 i (5 i)$ .

Reorder the factors in the terms $x (5 i)$ and $- 5 i x$ .

$y = (x \cdot x + x \cdot - 4 + 5 i x - 4 x - 4 \cdot - 4 - 4 (5 i) - 5 i x - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Subtract $5 i x$ from $5 i x$ .

$y = (x \cdot x + x \cdot - 4 + 0 - 4 x - 4 \cdot - 4 - 4 (5 i) - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Add $x \cdot x + x \cdot - 4$ and $0$ .

$y = (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 - 4 (5 i) - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $- 4$ by $- 4$ .

$y = (x^{2} - 4 x - 4 x + 16 - 4 (5 i) - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Multiply $5$ by $- 4$ .

$y = (x^{2} - 4 x - 4 x + 16 - 20 i - 5 i \cdot - 4 - 5 i (5 i)) (x + 1)$

Multiply $- 4$ by $- 5$ .

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 5 i (5 i)) (x + 1)$

Multiply $- 5 i (5 i)$ .

Multiply $5$ by $- 5$ .

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 i i) (x + 1)$

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

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 (i i)) (x + 1)$

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

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 (i i)) (x + 1)$

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

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 i^{1 + 1}) (x + 1)$

Add $1$ and $1$ .

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 i^{2}) (x + 1)$

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

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i - 25 \cdot - 1) (x + 1)$

Multiply $- 25$ by $- 1$ .

$y = (x^{2} - 4 x - 4 x + 16 - 20 i + 20 i + 25) (x + 1)$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 4 x - 4 x + 16 - 20 i + 20 i + 25$ .

Add $- 20 i$ and $20 i$ .

$y = (x^{2} - 4 x - 4 x + 16 + 0 + 25) (x + 1)$

Add $x^{2} - 4 x - 4 x + 16$ and $0$ .

$y = (x^{2} - 4 x - 4 x + 16 + 25) (x + 1)$

Subtract $4 x$ from $- 4 x$ .

$y = (x^{2} - 8 x + 16 + 25) (x + 1)$

Add $16$ and $25$ .

$y = (x^{2} - 8 x + 41) (x + 1)$

Expand $(x^{2} - 8 x + 41) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 1 - 8 x \cdot x - 8 x \cdot 1 + 41 x + 41 \cdot 1$

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 $1$ .

$y = x^{2} x + x^{2} \cdot 1 - 8 x \cdot x - 8 x \cdot 1 + 41 x + 41 \cdot 1$

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

$y = x^{2 + 1} + x^{2} \cdot 1 - 8 x \cdot x - 8 x \cdot 1 + 41 x + 41 \cdot 1$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot 1 - 8 x \cdot x - 8 x \cdot 1 + 41 x + 41 \cdot 1$

Multiply $x^{2}$ by $1$ .

$y = x^{3} + x^{2} - 8 x \cdot x - 8 x \cdot 1 + 41 x + 41 \cdot 1$

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

Move $x$ .

$y = x^{3} + x^{2} - 8 (x \cdot x) - 8 x \cdot 1 + 41 x + 41 \cdot 1$

Multiply $x$ by $x$ .

$y = x^{3} + x^{2} - 8 x^{2} - 8 x \cdot 1 + 41 x + 41 \cdot 1$

Multiply $- 8$ by $1$ .

$y = x^{3} + x^{2} - 8 x^{2} - 8 x + 41 x + 41 \cdot 1$

Multiply $41$ by $1$ .

$y = x^{3} + x^{2} - 8 x^{2} - 8 x + 41 x + 41$

Simplify by adding terms.

Subtract $8 x^{2}$ from $x^{2}$ .

$y = x^{3} - 7 x^{2} - 8 x + 41 x + 41$

Add $- 8 x$ and $41 x$ .

$y = x^{3} - 7 x^{2} + 33 x + 41$

Step 7

Find the Equation Given the Roots 4+5i , 4-5i , -1

Related Questions

Support LCMGCF.com by disabling your adblocker.