Find the Equation Given the Roots -1 , 3-2i

$- 1$ , $3 - 2 i$

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

The factor is $x + 1$

Step 3

The root at $x = 3 - 2 i$ was found by solving for $x$ when $x - (3 - 2 i) = y$ and $y = 0$ .

The factor is $x - 3 + 2 i$

Step 4

The root at $x = 3 + 2 i$ was found by solving for $x$ when $x - (3 + 2 i) = y$ and $y = 0$ .

The factor is $x - 3 - 2 i$

Step 5

Combine all the factors into a single equation.

$y = (x + 1) (x - 3 + 2 i) (x - 3 - 2 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 1) (x - 3 + 2 i) (x - 3 - 2 i)$ .

Expand $(x + 1) (x - 3 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 3 + x (2 i) + 1 x + 1 \cdot - 3 + 1 (2 i)) (x - 3 - 2 i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Move $2$ to the left of $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 3$ by $1$ .

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

Multiply $2 i$ by $1$ .

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

Add $- 3 x$ and $x$ .

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

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

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

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 $1$ .

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

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

Add $2$ and $1$ .

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

Move $- 3$ to the left of $x^{2}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 3$ by $2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 2 x i (- 2 i) - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $2 x i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x i i - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x (i i) - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x (i i) - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x i^{1 + 1} - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x i^{2} - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i - 4 x \cdot - 1 - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 1$ by $- 4$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x \cdot x - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

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

Move $x$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 (x \cdot x) - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} - 2 x \cdot - 3 - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 3$ by $- 2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x - 2 x (- 2 i) - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 2$ by $- 2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x - 3 \cdot - 3 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 3$ by $- 3$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 - 3 (- 2 i) + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 2$ by $- 3$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x + 2 i \cdot - 3 + 2 i (- 2 i)$

Multiply $- 3$ by $2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 i i$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 (i i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 (i i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 i^{2}$

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

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 4$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 3 x^{2} + x^{2} (- 2 i) + 2 x^{2} i - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 4$ .

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

$y = x^{3} - 3 x^{2} - 2 i x^{2} + 2 i x^{2} - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 4$

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

$y = x^{3} - 3 x^{2} + 0 - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 4$

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

$y = x^{3} - 3 x^{2} - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 6 i + 2 i x - 6 i + 4$

Subtract $6 i$ from $6 i$ .

$y = x^{3} - 3 x^{2} - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 2 i x + 0 + 4$

Add $x^{3} - 3 x^{2} - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 2 i x$ and $0$ .

$y = x^{3} - 3 x^{2} - 6 x i + 4 x - 2 x^{2} + 6 x + 4 x i - 3 x + 9 + 2 i x + 4$

Subtract $2 x^{2}$ from $- 3 x^{2}$ .

$y = x^{3} - 5 x^{2} - 6 x i + 4 x + 6 x + 4 x i - 3 x + 9 + 2 i x + 4$

Add $- 6 x i$ and $4 x i$ .

$y = x^{3} - 5 x^{2} - 2 x i + 4 x + 6 x - 3 x + 9 + 2 i x + 4$

Combine the opposite terms in $x^{3} - 5 x^{2} - 2 x i + 4 x + 6 x - 3 x + 9 + 2 i x + 4$ .

Reorder the factors in the terms $- 2 x i$ and $2 i x$ .

$y = x^{3} - 5 x^{2} - 2 i x + 4 x + 6 x - 3 x + 9 + 2 i x + 4$

Add $- 2 i x$ and $2 i x$ .

$y = x^{3} - 5 x^{2} + 4 x + 6 x - 3 x + 9 + 0 + 4$

Add $x^{3} - 5 x^{2} + 4 x + 6 x - 3 x + 9$ and $0$ .

$y = x^{3} - 5 x^{2} + 4 x + 6 x - 3 x + 9 + 4$

Add $4 x$ and $6 x$ .

$y = x^{3} - 5 x^{2} + 10 x - 3 x + 9 + 4$

Subtract $3 x$ from $10 x$ .

$y = x^{3} - 5 x^{2} + 7 x + 9 + 4$

Add $9$ and $4$ .

$y = x^{3} - 5 x^{2} + 7 x + 13$

Step 7

Find the Equation Given the Roots -1 , 3-2i

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