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

$1 + 3 i$ , $- 1$ , $2$

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 $x = 1 + 3 i$ was found by solving for $x$ when $x - (1 + 3 i) = y$ and $y = 0$ .

The factor is $x - 1 - 3 i$

Step 3

The root at $x = 1 - 3 i$ was found by solving for $x$ when $x - (1 - 3 i) = y$ and $y = 0$ .

The factor is $x - 1 + 3 i$

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

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

The factor is $x - 2$

Step 6

Combine all the factors into a single equation.

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

Step 7

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

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

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (3 i) - 1 x - 1 \cdot - 1 - 1 (3 i) - 3 i x - 3 i \cdot - 1 - 3 i (3 i)$ .

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

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

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

$y = (x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 (3 i) - 3 i \cdot - 1 - 3 i (3 i)) (x + 1) (x - 2)$

Add $x \cdot x + x \cdot - 1$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

Multiply $- 3 i (3 i)$ .

Multiply $3$ by $- 3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 9$ by $- 1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - x + 1 - 3 i + 3 i + 9$ .

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

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

Add $x^{2} - x - x + 1$ and $0$ .

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

Subtract $x$ from $- x$ .

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

Add $1$ and $9$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $1$ .

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

Multiply $10$ by $1$ .

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

Simplify by adding terms.

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

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

Add $- 2 x$ and $10 x$ .

$y = (x^{3} - x^{2} + 8 x + 10) (x - 2)$

Expand $(x^{3} - x^{2} + 8 x + 10) (x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 2 - x^{2} x - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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 $1$ .

$y = x^{3} x + x^{3} \cdot - 2 - x^{2} x - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

$y = x^{3 + 1} + x^{3} \cdot - 2 - x^{2} x - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 2 - x^{2} x - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

$y = x^{4} - 2 \cdot x^{3} - x^{2} x - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

Move $x$ .

$y = x^{4} - 2 x^{3} - (x \cdot x^{2}) - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

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

$y = x^{4} - 2 x^{3} - (x \cdot x^{2}) - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

$y = x^{4} - 2 x^{3} - x^{1 + 2} - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} - x^{3} - x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

Multiply $- 2$ by $- 1$ .

$y = x^{4} - 2 x^{3} - x^{3} + 2 x^{2} + 8 x \cdot x + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

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

Move $x$ .

$y = x^{4} - 2 x^{3} - x^{3} + 2 x^{2} + 8 (x \cdot x) + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} - x^{3} + 2 x^{2} + 8 x^{2} + 8 x \cdot - 2 + 10 x + 10 \cdot - 2$

Multiply $- 2$ by $8$ .

$y = x^{4} - 2 x^{3} - x^{3} + 2 x^{2} + 8 x^{2} - 16 x + 10 x + 10 \cdot - 2$

Multiply $10$ by $- 2$ .

$y = x^{4} - 2 x^{3} - x^{3} + 2 x^{2} + 8 x^{2} - 16 x + 10 x - 20$

Simplify by adding terms.

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

$y = x^{4} - 3 x^{3} + 2 x^{2} + 8 x^{2} - 16 x + 10 x - 20$

Add $2 x^{2}$ and $8 x^{2}$ .

$y = x^{4} - 3 x^{3} + 10 x^{2} - 16 x + 10 x - 20$

Add $- 16 x$ and $10 x$ .

$y = x^{4} - 3 x^{3} + 10 x^{2} - 6 x - 20$

Step 8

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

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