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

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

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 $x = 1 + 2 i$ was found by solving for $x$ when $x - (1 + 2 i) = y$ and $y = 0$ .

The factor is $x - 1 - 2 i$

Step 3

The root at $x = 1 - 2 i$ was found by solving for $x$ when $x - (1 - 2 i) = y$ and $y = 0$ .

The factor is $x - 1 + 2 i$

Step 4

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

The factor is $x + 2 + i$

Step 5

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

The factor is $x + 2 - i$

Step 6

Combine all the factors into a single equation.

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

Step 7

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

Expand $(x - 1 - 2 i) (x - 1 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (2 i) - 1 x - 1 \cdot - 1 - 1 (2 i) - 2 i x - 2 i \cdot - 1 - 2 i (2 i)$ .

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

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

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

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

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 2$ .

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

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - x + 1 - 2 i + 2 i + 4$ .

Add $- 2 i$ and $2 i$ .

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

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

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

Subtract $x$ from $- x$ .

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

Add $1$ and $4$ .

$y = (x^{2} - 2 x + 5) (x + 2 + i) (x + 2 - i)$

Expand $(x^{2} - 2 x + 5) (x + 2 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot 2 + x^{2} i - 2 x \cdot x - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot 2 + x^{2} i - 2 x \cdot x - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

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

$y = (x^{2 + 1} + x^{2} \cdot 2 + x^{2} i - 2 x \cdot x - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot 2 + x^{2} i - 2 x \cdot x - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

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

$y = (x^{3} + 2 \cdot x^{2} + x^{2} i - 2 x \cdot x - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

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

Move $x$ .

$y = (x^{3} + 2 x^{2} + x^{2} i - 2 (x \cdot x) - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

Multiply $x$ by $x$ .

$y = (x^{3} + 2 x^{2} + x^{2} i - 2 x^{2} - 2 x \cdot 2 - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

Multiply $2$ by $- 2$ .

$y = (x^{3} + 2 x^{2} + x^{2} i - 2 x^{2} - 4 x - 2 x i + 5 x + 5 \cdot 2 + 5 i) (x + 2 - i)$

Multiply $5$ by $2$ .

$y = (x^{3} + 2 x^{2} + x^{2} i - 2 x^{2} - 4 x - 2 x i + 5 x + 10 + 5 i) (x + 2 - i)$

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 2 x^{2} + x^{2} i - 2 x^{2} - 4 x - 2 x i + 5 x + 10 + 5 i$ .

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

$y = (x^{3} + x^{2} i + 0 - 4 x - 2 x i + 5 x + 10 + 5 i) (x + 2 - i)$

Add $x^{3} + x^{2} i$ and $0$ .

$y = (x^{3} + x^{2} i - 4 x - 2 x i + 5 x + 10 + 5 i) (x + 2 - i)$

Add $- 4 x$ and $5 x$ .

$y = (x^{3} + x^{2} i - 2 x i + x + 10 + 5 i) (x + 2 - i)$

Expand $(x^{3} + x^{2} i - 2 x i + x + 10 + 5 i) (x + 2 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot 2 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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 $1$ .

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

$y = x^{3 + 1} + x^{3} \cdot 2 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot 2 + x^{3} (- i) + x^{2} i x + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 \cdot x^{3} + x^{3} (- i) + x^{2} i x + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

Move $x$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x \cdot x^{2} i + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

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

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{1 + 2} i + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Add $1$ and $2$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + x^{2} i \cdot 2 + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 \cdot (x^{2} i) + x^{2} i (- i) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} (- (i i)) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} (- i^{1 + 1}) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Add $1$ and $1$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} (- i^{2}) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} (1) - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} \cdot 1 - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x i x - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

Move $x$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 (x \cdot x) i - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 2 x i \cdot 2 - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $2$ by $- 2$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x i (- i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $- 2 x i (- i)$ .

Multiply $- 1$ by $- 2$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x i i + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x (i i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x (i i) + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x i^{1 + 1} + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Add $1$ and $1$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x i^{2} + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i + 2 x \cdot - 1 + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $- 1$ by $2$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x \cdot x + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + x \cdot 2 + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 \cdot x + x (- i) + 10 x + 10 \cdot 2 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $10$ by $2$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 + 10 (- i) + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $- 1$ by $10$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 5 i \cdot 2 + 5 i (- i)$

Multiply $2$ by $5$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5 i (- i)$

Multiply $5 i (- i)$ .

Multiply $- 1$ by $5$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 i i$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 (i i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 (i i)$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 i^{2}$

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

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i - 5 \cdot - 1$

Multiply $- 5$ by $- 1$ .

$y = x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + 2 x^{3} + x^{3} (- i) + x^{3} i + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$ .

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

$y = x^{4} + 2 x^{3} - i x^{3} + i x^{3} + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + 0 + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + 2 x^{2} i + x^{2} - 2 x^{2} i - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + x^{2} + 0 - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + x^{2} - 4 x i - 2 x + x^{2} + 2 x + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + x^{2} - 4 x i + x^{2} + 0 + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

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

$y = x^{4} + 2 x^{3} + x^{2} - 4 x i + x^{2} + x (- i) + 10 x + 20 - 10 i + 5 i x + 10 i + 5$

Add $- 10 i$ and $10 i$ .

$y = x^{4} + 2 x^{3} + x^{2} - 4 x i + x^{2} + x (- i) + 10 x + 20 + 5 i x + 0 + 5$

Add $x^{4} + 2 x^{3} + x^{2} - 4 x i + x^{2} + x (- i) + 10 x + 20 + 5 i x$ and $0$ .

$y = x^{4} + 2 x^{3} + x^{2} - 4 x i + x^{2} + x (- i) + 10 x + 20 + 5 i x + 5$

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

$y = x^{4} + 2 x^{3} + 2 x^{2} - 4 x i + x (- i) + 10 x + 20 + 5 i x + 5$

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

Reorder $x$ and $- 1$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} - 4 x i - 1 \cdot (x i) + 10 x + 20 + 5 i x + 5$

Subtract $x i$ from $- 4 x i$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} - 5 x i + 10 x + 20 + 5 i x + 5$

Combine the opposite terms in $x^{4} + 2 x^{3} + 2 x^{2} - 5 x i + 10 x + 20 + 5 i x + 5$ .

Reorder the factors in the terms $- 5 x i$ and $5 i x$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} - 5 i x + 10 x + 20 + 5 i x + 5$

Add $- 5 i x$ and $5 i x$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} + 10 x + 20 + 0 + 5$

Add $x^{4} + 2 x^{3} + 2 x^{2} + 10 x + 20$ and $0$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} + 10 x + 20 + 5$

Add $20$ and $5$ .

$y = x^{4} + 2 x^{3} + 2 x^{2} + 10 x + 25$

Step 8

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

Related Questions

Support LCMGCF.com by disabling your adblocker.