Find the Equation Given the Roots 5+3i , 5-3i , -1

$5 + 3 i$ , $5 - 3 i$ , $- 1$

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 $x = 5 + 3 i$ was found by solving for $x$ when $x - (5 + 3 i) = y$ and $y = 0$ .

The factor is $x - 5 - 3 i$

Step 3

The root at $x = 5 - 3 i$ was found by solving for $x$ when $x - (5 - 3 i) = y$ and $y = 0$ .

The factor is $x - 5 + 3 i$

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.

$y = (x - 5 - 3 i) (x - 5 + 3 i) (x + 1)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 5 - 3 i) (x - 5 + 3 i) (x + 1)$ .

Expand $(x - 5 - 3 i) (x - 5 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 5 + x (3 i) - 5 x - 5 \cdot - 5 - 5 (3 i) - 3 i x - 3 i \cdot - 5 - 3 i (3 i)) (x + 1)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 5 + x (3 i) - 5 x - 5 \cdot - 5 - 5 (3 i) - 3 i x - 3 i \cdot - 5 - 3 i (3 i)$ .

Reorder the factors in the terms $x (3 i)$ and $- 3 i x$ .

$y = (x \cdot x + x \cdot - 5 + 3 i x - 5 x - 5 \cdot - 5 - 5 (3 i) - 3 i x - 3 i \cdot - 5 - 3 i (3 i)) (x + 1)$

Subtract $3 i x$ from $3 i x$ .

$y = (x \cdot x + x \cdot - 5 + 0 - 5 x - 5 \cdot - 5 - 5 (3 i) - 3 i \cdot - 5 - 3 i (3 i)) (x + 1)$

Add $x \cdot x + x \cdot - 5$ and $0$ .

$y = (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (3 i) - 3 i \cdot - 5 - 3 i (3 i)) (x + 1)$

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 5$ to the left of $x$ .

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

Multiply $- 5$ by $- 5$ .

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

Multiply $3$ by $- 5$ .

$y = (x^{2} - 5 x - 5 x + 25 - 15 i - 3 i \cdot - 5 - 3 i (3 i)) (x + 1)$

Multiply $- 5$ by $- 3$ .

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 3 i (3 i)) (x + 1)$

Multiply $- 3 i (3 i)$ .

Multiply $3$ by $- 3$ .

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 i i) (x + 1)$

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

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 (i i)) (x + 1)$

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

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 (i i)) (x + 1)$

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

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 i^{1 + 1}) (x + 1)$

Add $1$ and $1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 i^{2}) (x + 1)$

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

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i - 9 \cdot - 1) (x + 1)$

Multiply $- 9$ by $- 1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 15 i + 15 i + 9) (x + 1)$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 5 x - 5 x + 25 - 15 i + 15 i + 9$ .

Add $- 15 i$ and $15 i$ .

$y = (x^{2} - 5 x - 5 x + 25 + 0 + 9) (x + 1)$

Add $x^{2} - 5 x - 5 x + 25$ and $0$ .

$y = (x^{2} - 5 x - 5 x + 25 + 9) (x + 1)$

Subtract $5 x$ from $- 5 x$ .

$y = (x^{2} - 10 x + 25 + 9) (x + 1)$

Add $25$ and $9$ .

$y = (x^{2} - 10 x + 34) (x + 1)$

Expand $(x^{2} - 10 x + 34) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 34 x + 34 \cdot 1$

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 $1$ .

$y = x^{2} x + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 34 x + 34 \cdot 1$

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

$y = x^{2 + 1} + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 34 x + 34 \cdot 1$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 34 x + 34 \cdot 1$

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

$y = x^{3} + x^{2} - 10 x \cdot x - 10 x \cdot 1 + 34 x + 34 \cdot 1$

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

Move $x$ .

$y = x^{3} + x^{2} - 10 (x \cdot x) - 10 x \cdot 1 + 34 x + 34 \cdot 1$

Multiply $x$ by $x$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x \cdot 1 + 34 x + 34 \cdot 1$

Multiply $- 10$ by $1$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x + 34 x + 34 \cdot 1$

Multiply $34$ by $1$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x + 34 x + 34$

Simplify by adding terms.

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

$y = x^{3} - 9 x^{2} - 10 x + 34 x + 34$

Add $- 10 x$ and $34 x$ .

$y = x^{3} - 9 x^{2} + 24 x + 34$

Step 7

Find the Equation Given the Roots 5+3i , 5-3i , -1

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