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 $x = 4 - 3 i$ was found by solving for $x$ when $x - (4 - 3 i) = y$ and $y = 0$ .

The factor is $x - 4 + 3 i$

Step 4

The root at $x = 4 + 3 i$ was found by solving for $x$ when $x - (4 + 3 i) = y$ and $y = 0$ .

The factor is $x - 4 - 3 i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

Expand $(x + 1) (x - 4 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $x$ by $1$ .

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

Multiply $- 4$ by $1$ .

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

Multiply $3 i$ by $1$ .

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

Add $- 4 x$ and $x$ .

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

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

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

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.

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

Add $2$ and $1$ .

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

Move $- 4$ to the left of $x^{2}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4$ by $3$ .

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

Multiply $3 x i (- 3 i)$ .

Multiply $- 3$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 9$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4$ by $- 3$ .

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

Multiply $- 3$ by $- 3$ .

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

Multiply $- 4$ by $- 4$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 - 4 (- 3 i) + 3 i x + 3 i \cdot - 4 + 3 i (- 3 i)$

Multiply $- 3$ by $- 4$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i (- 3 i)$

Multiply $- 4$ by $3$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 3 i (- 3 i)$

Multiply $3 i (- 3 i)$ .

Multiply $- 3$ by $3$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 i i$

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

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 (i i)$

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

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 (i i)$

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

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 i^{2}$

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

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i - 9 \cdot - 1$

Multiply $- 9$ by $- 1$ .

$y = x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 9$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 4 x^{2} + x^{2} (- 3 i) + 3 x^{2} i - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 9$ .

Reorder the factors in the terms $x^{2} (- 3 i)$ and $3 x^{2} i$ .

$y = x^{3} - 4 x^{2} - 3 i x^{2} + 3 i x^{2} - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 9$

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

$y = x^{3} - 4 x^{2} + 0 - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 9$

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

$y = x^{3} - 4 x^{2} - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 12 i + 3 i x - 12 i + 9$

Subtract $12 i$ from $12 i$ .

$y = x^{3} - 4 x^{2} - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 3 i x + 0 + 9$

Add $x^{3} - 4 x^{2} - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 3 i x$ and $0$ .

$y = x^{3} - 4 x^{2} - 12 x i + 9 x - 3 x^{2} + 12 x + 9 x i - 4 x + 16 + 3 i x + 9$

Subtract $3 x^{2}$ from $- 4 x^{2}$ .

$y = x^{3} - 7 x^{2} - 12 x i + 9 x + 12 x + 9 x i - 4 x + 16 + 3 i x + 9$

Add $- 12 x i$ and $9 x i$ .

$y = x^{3} - 7 x^{2} - 3 x i + 9 x + 12 x - 4 x + 16 + 3 i x + 9$

Combine the opposite terms in $x^{3} - 7 x^{2} - 3 x i + 9 x + 12 x - 4 x + 16 + 3 i x + 9$ .

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

$y = x^{3} - 7 x^{2} - 3 i x + 9 x + 12 x - 4 x + 16 + 3 i x + 9$

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

$y = x^{3} - 7 x^{2} + 9 x + 12 x - 4 x + 16 + 0 + 9$

Add $x^{3} - 7 x^{2} + 9 x + 12 x - 4 x + 16$ and $0$ .

$y = x^{3} - 7 x^{2} + 9 x + 12 x - 4 x + 16 + 9$

Add $9 x$ and $12 x$ .

$y = x^{3} - 7 x^{2} + 21 x - 4 x + 16 + 9$

Subtract $4 x$ from $21 x$ .

$y = x^{3} - 7 x^{2} + 17 x + 16 + 9$

Add $16$ and $9$ .

$y = x^{3} - 7 x^{2} + 17 x + 25$

Step 7

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

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