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

$- 1$ , $1 - i$

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

The factor is $x - 1 + i$

Step 4

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

The factor is $x - 1 - i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $i$ by $1$ .

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

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

Add $- x$ and $x$ .

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

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

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

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

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

Simplify terms.

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

Reorder the factors in the terms $x i \cdot - 1$ and $i x$ .

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

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

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

Add $x^{2} x + x^{2} \cdot - 1 + x^{2} (- i) + x i x$ and $0$ .

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

Move $- 1$ to the left of $x^{2}$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x i (- i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $i$ by $1$ .

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

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

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

Rewrite $- 1 i$ as $- i$ .

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

Multiply $i (- i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Simplify by adding terms.

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

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

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

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

$y = x^{3} - x^{2} + 0 + x - x + 1 + i - i + 1$

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

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

Subtract $x$ from $x$ .

$y = x^{3} - x^{2} + 0 + 1 + i - i + 1$

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

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

Subtract $i$ from $i$ .

$y = x^{3} - x^{2} + 1 + 0 + 1$

Add $x^{3} - x^{2} + 1$ and $0$ .

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

Add $1$ and $1$ .

$y = x^{3} - x^{2} + 2$

Step 7

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

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