Find the Equation Given the Roots -4 , 3-i

$- 4$ , $3 - i$

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

The factor is $x - 3 + i$

Step 4

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

The factor is $x - 3 - i$

Step 5

Combine all the factors into a single equation.

$y = (x + 4) (x - 3 + i) (x - 3 - i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 4) (x - 3 + i) (x - 3 - i)$ .

Expand $(x + 4) (x - 3 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 3 + x i + 4 x + 4 \cdot - 3 + 4 i) (x - 3 - i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $4$ by $- 3$ .

$y = (x^{2} - 3 x + x i + 4 x - 12 + 4 i) (x - 3 - i)$

Add $- 3 x$ and $4 x$ .

$y = (x^{2} + x i + x - 12 + 4 i) (x - 3 - i)$

Expand $(x^{2} + x i + x - 12 + 4 i) (x - 3 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 3 + x^{2} (- i) + x i x + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 3 + x^{2} (- i) + x i x + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{2 + 1} + x^{2} \cdot - 3 + x^{2} (- i) + x i x + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 3 + x^{2} (- i) + x i x + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 \cdot x^{2} + x^{2} (- i) + x i x + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

Move $x$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x \cdot x i + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i + x i \cdot - 3 + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 \cdot (x i) + x i (- i) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $x i (- i)$ .

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x (- (i i)) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x (- (i i)) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x (- i^{1 + 1}) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x (- i^{2}) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x (1) + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x \cdot 1 + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $x$ by $1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x \cdot x + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} + x \cdot - 3 + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 \cdot x + x (- i) - 12 x - 12 \cdot - 3 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $- 12$ by $- 3$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 - 12 (- i) + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $- 1$ by $- 12$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x + 4 i \cdot - 3 + 4 i (- i)$

Multiply $- 3$ by $4$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4 i (- i)$

Multiply $4 i (- i)$ .

Multiply $- 1$ by $4$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 i i$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 (i i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 (i i)$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 i^{2}$

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

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$y = x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 3 x^{2} + x^{2} (- i) + x^{2} i - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4$ .

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

$y = x^{3} - 3 x^{2} - i x^{2} + i x^{2} - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4$

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

$y = x^{3} - 3 x^{2} + 0 - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4$

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

$y = x^{3} - 3 x^{2} - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 12 i + 4 i x - 12 i + 4$

Subtract $12 i$ from $12 i$ .

$y = x^{3} - 3 x^{2} - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 4 i x + 0 + 4$

Add $x^{3} - 3 x^{2} - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 4 i x$ and $0$ .

$y = x^{3} - 3 x^{2} - 3 x i + x + x^{2} - 3 x + x (- i) - 12 x + 36 + 4 i x + 4$

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

$y = x^{3} - 2 x^{2} - 3 x i + x - 3 x + x (- i) - 12 x + 36 + 4 i x + 4$

Add $- 3 x i$ and $x (- i)$ .

Reorder $x$ and $- 1$ .

$y = x^{3} - 2 x^{2} - 3 x i - 1 \cdot (x i) + x - 3 x - 12 x + 36 + 4 i x + 4$

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

$y = x^{3} - 2 x^{2} - 4 x i + x - 3 x - 12 x + 36 + 4 i x + 4$

Combine the opposite terms in $x^{3} - 2 x^{2} - 4 x i + x - 3 x - 12 x + 36 + 4 i x + 4$ .

Reorder the factors in the terms $- 4 x i$ and $4 i x$ .

$y = x^{3} - 2 x^{2} - 4 i x + x - 3 x - 12 x + 36 + 4 i x + 4$

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

$y = x^{3} - 2 x^{2} + x - 3 x - 12 x + 36 + 0 + 4$

Add $x^{3} - 2 x^{2} + x - 3 x - 12 x + 36$ and $0$ .

$y = x^{3} - 2 x^{2} + x - 3 x - 12 x + 36 + 4$

Subtract $3 x$ from $x$ .

$y = x^{3} - 2 x^{2} - 2 x - 12 x + 36 + 4$

Subtract $12 x$ from $- 2 x$ .

$y = x^{3} - 2 x^{2} - 14 x + 36 + 4$

Add $36$ and $4$ .

$y = x^{3} - 2 x^{2} - 14 x + 40$

Step 7

Find the Equation Given the Roots -4 , 3-i

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