Find the Equation Given the Roots 2 , 5-i

$2$ , $5 - i$

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

The factor is $x - 2$

Step 3

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

The factor is $x - 5 + i$

Step 4

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

The factor is $x - 5 - i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 5$ .

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

Subtract $2 x$ from $- 5 x$ .

$y = (x^{2} + x i - 7 x + 10 - 2 i) (x - 5 - i)$

Expand $(x^{2} + x i - 7 x + 10 - 2 i) (x - 5 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 5 + x^{2} (- i) + x i x + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 5 + x^{2} (- i) + x i x + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

$y = x^{2 + 1} + x^{2} \cdot - 5 + x^{2} (- i) + x i x + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 5 + x^{2} (- i) + x i x + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Move $- 5$ to the left of $x^{2}$ .

$y = x^{3} - 5 \cdot x^{2} + x^{2} (- i) + x i x + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

Move $x$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x \cdot x i + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i + x i \cdot - 5 + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 \cdot (x i) + x i (- i) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $x i (- i)$ .

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x (- i^{1 + 1}) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Add $1$ and $1$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x (- i^{2}) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x (1) - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x \cdot 1 - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $x$ by $1$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x \cdot x - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

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

Move $x$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 (x \cdot x) - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} - 7 x \cdot - 5 - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $- 5$ by $- 7$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x - 7 x (- i) + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $- 1$ by $- 7$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x + 10 \cdot - 5 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $10$ by $- 5$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 + 10 (- i) - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $- 1$ by $10$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x - 2 i \cdot - 5 - 2 i (- i)$

Multiply $- 5$ by $- 2$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2 i (- i)$

Multiply $- 2 i (- i)$ .

Multiply $- 1$ by $- 2$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 i i$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 (i i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 (i i)$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 i^{2}$

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

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$y = x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 5 x^{2} + x^{2} (- i) + x^{2} i - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2$ .

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

$y = x^{3} - 5 x^{2} - i x^{2} + i x^{2} - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2$

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

$y = x^{3} - 5 x^{2} + 0 - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2$

Add $x^{3} - 5 x^{2}$ and $0$ .

$y = x^{3} - 5 x^{2} - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 10 i - 2 i x + 10 i - 2$

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

$y = x^{3} - 5 x^{2} - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 2 i x + 0 - 2$

Add $x^{3} - 5 x^{2} - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 2 i x$ and $0$ .

$y = x^{3} - 5 x^{2} - 5 x i + x - 7 x^{2} + 35 x + 7 x i + 10 x - 50 - 2 i x - 2$

Subtract $7 x^{2}$ from $- 5 x^{2}$ .

$y = x^{3} - 12 x^{2} - 5 x i + x + 35 x + 7 x i + 10 x - 50 - 2 i x - 2$

Add $- 5 x i$ and $7 x i$ .

$y = x^{3} - 12 x^{2} + 2 x i + x + 35 x + 10 x - 50 - 2 i x - 2$

Combine the opposite terms in $x^{3} - 12 x^{2} + 2 x i + x + 35 x + 10 x - 50 - 2 i x - 2$ .

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

$y = x^{3} - 12 x^{2} + 2 i x + x + 35 x + 10 x - 50 - 2 i x - 2$

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

$y = x^{3} - 12 x^{2} + x + 35 x + 10 x - 50 + 0 - 2$

Add $x^{3} - 12 x^{2} + x + 35 x + 10 x - 50$ and $0$ .

$y = x^{3} - 12 x^{2} + x + 35 x + 10 x - 50 - 2$

Add $x$ and $35 x$ .

$y = x^{3} - 12 x^{2} + 36 x + 10 x - 50 - 2$

Add $36 x$ and $10 x$ .

$y = x^{3} - 12 x^{2} + 46 x - 50 - 2$

Subtract $2$ from $- 50$ .

$y = x^{3} - 12 x^{2} + 46 x - 52$

Step 7

Find the Equation Given the Roots 2 , 5-i

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