Find the Equation Given the Roots -1 , 2-5i

$- 1$ , $2 - 5 i$

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 $x = 2 - 5 i$ was found by solving for $x$ when $x - (2 - 5 i) = y$ and $y = 0$ .

The factor is $x - 2 + 5 i$

Step 4

The root at $x = 2 + 5 i$ was found by solving for $x$ when $x - (2 + 5 i) = y$ and $y = 0$ .

The factor is $x - 2 - 5 i$

Step 5

Combine all the factors into a single equation.

$y = (x + 1) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 1) (x - 2 + 5 i) (x - 2 - 5 i)$ .

Expand $(x + 1) (x - 2 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Move $5$ to the left of $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 2$ by $1$ .

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

Multiply $5 i$ by $1$ .

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

Add $- 2 x$ and $x$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $5$ .

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

Multiply $5 x i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x i i - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x (i i) - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x (i i) - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x i^{1 + 1} - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Add $1$ and $1$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x i^{2} - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i - 25 x \cdot - 1 - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 1$ by $- 25$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x \cdot x - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

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

Move $x$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - (x \cdot x) - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} - x \cdot - 2 - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 2$ by $- 1$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x - x (- 5 i) - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 5$ by $- 1$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x - 2 \cdot - 2 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 2$ by $- 2$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 - 2 (- 5 i) + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 5$ by $- 2$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x + 5 i \cdot - 2 + 5 i (- 5 i)$

Multiply $- 2$ by $5$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 5 i (- 5 i)$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 i i$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 (i i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 (i i)$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 i^{2}$

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

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i - 25 \cdot - 1$

Multiply $- 25$ by $- 1$ .

$y = x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 25$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 2 x^{2} + x^{2} (- 5 i) + 5 x^{2} i - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 25$ .

Reorder the factors in the terms $x^{2} (- 5 i)$ and $5 x^{2} i$ .

$y = x^{3} - 2 x^{2} - 5 i x^{2} + 5 i x^{2} - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 25$

Add $- 5 i x^{2}$ and $5 i x^{2}$ .

$y = x^{3} - 2 x^{2} + 0 - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 25$

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

$y = x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 2 x + 5 x i - 2 x + 4 + 10 i + 5 i x - 10 i + 25$

Subtract $2 x$ from $2 x$ .

$y = x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i + 0 + 4 + 10 i + 5 i x - 10 i + 25$

Add $x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i$ and $0$ .

$y = x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i + 4 + 10 i + 5 i x - 10 i + 25$

Subtract $10 i$ from $10 i$ .

$y = x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i + 4 + 5 i x + 0 + 25$

Add $x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i + 4 + 5 i x$ and $0$ .

$y = x^{3} - 2 x^{2} - 10 x i + 25 x - x^{2} + 5 x i + 4 + 5 i x + 25$

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

$y = x^{3} - 3 x^{2} - 10 x i + 25 x + 5 x i + 4 + 5 i x + 25$

Add $- 10 x i$ and $5 x i$ .

$y = x^{3} - 3 x^{2} - 5 x i + 25 x + 4 + 5 i x + 25$

Combine the opposite terms in $x^{3} - 3 x^{2} - 5 x i + 25 x + 4 + 5 i x + 25$ .

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

$y = x^{3} - 3 x^{2} - 5 i x + 25 x + 4 + 5 i x + 25$

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

$y = x^{3} - 3 x^{2} + 25 x + 4 + 0 + 25$

Add $x^{3} - 3 x^{2} + 25 x + 4$ and $0$ .

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

Add $4$ and $25$ .

$y = x^{3} - 3 x^{2} + 25 x + 29$

Step 7

Find the Equation Given the Roots -1 , 2-5i

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