Find the Equation Given the Roots -1 , 4 , 3i

$- 1$ , $4$ , $3 i$

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

The factor is $x + 1$

Step 3

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

The factor is $x - 4$

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

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 6

Combine all the factors into a single equation.

$y = (x + 1) (x - 4) (x - 3 i) (x + 3 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x + 1) (x - 4) (x - 3 i) (x + 3 i)$ .

Expand $(x + 1) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 4) + 1 (x - 4)) (x - 3 i) (x + 3 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 4 + 1 (x - 4)) (x - 3 i) (x + 3 i)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 4 + 1 x + 1 \cdot - 4) (x - 3 i) (x + 3 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $x$ by $1$ .

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

Multiply $- 4$ by $1$ .

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

Add $- 4 x$ and $x$ .

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

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

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

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

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

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

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

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

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

Add $2$ and $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 3$ by $- 3$ .

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

Multiply $- 3$ by $- 4$ .

$y = (x^{3} + x^{2} (- 3 i) - 3 x^{2} + 9 x i - 4 x + 12 i) (x + 3 i)$

Expand $(x^{3} + x^{2} (- 3 i) - 3 x^{2} + 9 x i - 4 x + 12 i) (x + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (3 i) + x^{2} (- 3 i) x + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$y = x^{3} x + x^{3} (3 i) + x^{2} (- 3 i) x + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{3 + 1} + x^{3} (3 i) + x^{2} (- 3 i) x + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (3 i) + x^{2} (- 3 i) x + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 \cdot (x^{3} i) + x^{2} (- 3 i) x + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{4} + 3 x^{3} i + x \cdot x^{2} (- 3 i) + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

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

$y = x^{4} + 3 x^{3} i + x \cdot x^{2} (- 3 i) + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{1 + 2} (- 3 i) + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Add $1$ and $2$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 3 i) (3 i) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $x^{2} (- 3 i) (3 i)$ .

Multiply $3$ by $- 3$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 i) i - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 (i i)) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 (i i)) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 i^{1 + 1}) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Add $1$ and $1$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 i^{2}) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} (- 9 \cdot - 1) - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $- 9$ by $- 1$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + x^{2} \cdot 9 - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Move $9$ to the left of $x^{2}$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 \cdot x^{2} - 3 x^{2} x - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 (x \cdot x^{2}) - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 (x \cdot x^{2}) - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{1 + 2} - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Add $1$ and $2$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 3 x^{2} (3 i) + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $3$ by $- 3$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x i x + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

Move $x$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 (x \cdot x) i + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 9 x i (3 i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $9 x i (3 i)$ .

Multiply $3$ by $9$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x i i - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x (i i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x (i i) - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x i^{1 + 1} - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Add $1$ and $1$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x i^{2} - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i + 27 x \cdot - 1 - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $- 1$ by $27$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x \cdot x - 4 x (3 i) + 12 i x + 12 i (3 i)$

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

Move $x$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 (x \cdot x) - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $x$ by $x$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 4 x (3 i) + 12 i x + 12 i (3 i)$

Multiply $3$ by $- 4$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 12 i (3 i)$

Multiply $12 i (3 i)$ .

Multiply $3$ by $12$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 i i$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 (i i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 (i i)$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 i^{2}$

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

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x + 36 \cdot - 1$

Multiply $36$ by $- 1$ .

$y = x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + 3 x^{3} i + x^{3} (- 3 i) + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$ .

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

$y = x^{4} + 3 i x^{3} - 3 i x^{3} + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

Subtract $3 i x^{3}$ from $3 i x^{3}$ .

$y = x^{4} + 0 + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

Add $x^{4}$ and $0$ .

$y = x^{4} + 9 x^{2} - 3 x^{3} - 9 x^{2} i + 9 x^{2} i - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

Add $- 9 x^{2} i$ and $9 x^{2} i$ .

$y = x^{4} + 9 x^{2} - 3 x^{3} + 0 - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

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

$y = x^{4} + 9 x^{2} - 3 x^{3} - 27 x - 4 x^{2} - 12 x i + 12 i x - 36$

Reorder the factors in the terms $- 12 x i$ and $12 i x$ .

$y = x^{4} + 9 x^{2} - 3 x^{3} - 27 x - 4 x^{2} - 12 i x + 12 i x - 36$

Add $- 12 i x$ and $12 i x$ .

$y = x^{4} + 9 x^{2} - 3 x^{3} - 27 x - 4 x^{2} + 0 - 36$

Add $x^{4} + 9 x^{2} - 3 x^{3} - 27 x - 4 x^{2}$ and $0$ .

$y = x^{4} + 9 x^{2} - 3 x^{3} - 27 x - 4 x^{2} - 36$

Subtract $4 x^{2}$ from $9 x^{2}$ .

$y = x^{4} + 5 x^{2} - 3 x^{3} - 27 x - 36$

Move $5 x^{2}$ .

$y = x^{4} - 3 x^{3} + 5 x^{2} - 27 x - 36$

Step 8

Find the Equation Given the Roots -1 , 4 , 3i

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