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

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

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

The factor is $x - 2 - 2 i$

Step 3

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

The factor is $x - 2 + 2 i$

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.

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

Step 7

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

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

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 2 + x (2 i) - 2 x - 2 \cdot - 2 - 2 (2 i) - 2 i x - 2 i \cdot - 2 - 2 i (2 i)$ .

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

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

Subtract $2 i x$ from $2 i x$ .

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

Add $x \cdot x + x \cdot - 2$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 2$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Simplify by adding terms.

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

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

$y = (x^{2} - 2 x - 2 x + 4 + 0 + 4) (x + 1) (x - 2)$

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

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

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

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

Add $4$ and $4$ .

$y = (x^{2} - 4 x + 8) (x + 1) (x - 2)$

Expand $(x^{2} - 4 x + 8) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot 1 - 4 x \cdot x - 4 x \cdot 1 + 8 x + 8 \cdot 1) (x - 2)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot 1 - 4 x \cdot x - 4 x \cdot 1 + 8 x + 8 \cdot 1) (x - 2)$

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

$y = (x^{2 + 1} + x^{2} \cdot 1 - 4 x \cdot x - 4 x \cdot 1 + 8 x + 8 \cdot 1) (x - 2)$

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4$ by $1$ .

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

Multiply $8$ by $1$ .

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

Simplify by adding terms.

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

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

Add $- 4 x$ and $8 x$ .

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

Expand $(x^{3} - 3 x^{2} + 4 x + 8) (x - 2)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $1$ .

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

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

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

Add $3$ and $1$ .

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

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

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 2$ by $- 3$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x \cdot x + 4 x \cdot - 2 + 8 x + 8 \cdot - 2$

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

Move $x$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 (x \cdot x) + 4 x \cdot - 2 + 8 x + 8 \cdot - 2$

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} + 4 x \cdot - 2 + 8 x + 8 \cdot - 2$

Multiply $- 2$ by $4$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} - 8 x + 8 x + 8 \cdot - 2$

Multiply $8$ by $- 2$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} - 8 x + 8 x - 16$

Simplify by adding terms.

Combine the opposite terms in $x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} - 8 x + 8 x - 16$ .

Add $- 8 x$ and $8 x$ .

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} + 0 - 16$

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

$y = x^{4} - 2 x^{3} - 3 x^{3} + 6 x^{2} + 4 x^{2} - 16$

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

$y = x^{4} - 5 x^{3} + 6 x^{2} + 4 x^{2} - 16$

Add $6 x^{2}$ and $4 x^{2}$ .

$y = x^{4} - 5 x^{3} + 10 x^{2} - 16$

Step 8

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

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