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

$- 5 + 2 i$ , $- 5 i$

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

The factor is $x + 5 - 2 i$

Step 3

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

The factor is $x + 5 + 2 i$

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

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 6

Combine all the factors into a single equation.

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

Step 7

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

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

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot 5 + x (2 i) + 5 x + 5 \cdot 5 + 5 (2 i) - 2 i x - 2 i \cdot 5 - 2 i (2 i)$ .

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

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

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

$y = (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 + 5 (2 i) + 0 - 2 i \cdot 5 - 2 i (2 i)) (x + 5 i) (x - 5 i)$

Add $x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 + 5 (2 i)$ and $0$ .

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

Reorder the factors in the terms $5 (2 i)$ and $- 2 i \cdot 5$ .

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

Subtract $2 \cdot 5 i$ from $2 \cdot 5 i$ .

$y = (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 + 0 - 2 i (2 i)) (x + 5 i) (x - 5 i)$

Add $x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $5$ by $5$ .

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

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Simplify by adding terms.

Add $5 x$ and $5 x$ .

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

Add $25$ and $4$ .

$y = (x^{2} + 10 x + 29) (x + 5 i) (x - 5 i)$

Expand $(x^{2} + 10 x + 29) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} (5 i) + 10 x \cdot x + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

Simplify each term.

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

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

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

$y = (x^{2} x + x^{2} (5 i) + 10 x \cdot x + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

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

$y = (x^{2 + 1} + x^{2} (5 i) + 10 x \cdot x + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} (5 i) + 10 x \cdot x + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

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

$y = (x^{3} + 5 \cdot (x^{2} i) + 10 x \cdot x + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

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

Move $x$ .

$y = (x^{3} + 5 x^{2} i + 10 (x \cdot x) + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

Multiply $x$ by $x$ .

$y = (x^{3} + 5 x^{2} i + 10 x^{2} + 10 x (5 i) + 29 x + 29 (5 i)) (x - 5 i)$

Multiply $5$ by $10$ .

$y = (x^{3} + 5 x^{2} i + 10 x^{2} + 50 x i + 29 x + 29 (5 i)) (x - 5 i)$

Multiply $5$ by $29$ .

$y = (x^{3} + 5 x^{2} i + 10 x^{2} + 50 x i + 29 x + 145 i) (x - 5 i)$

Expand $(x^{3} + 5 x^{2} i + 10 x^{2} + 50 x i + 29 x + 145 i) (x - 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} (- 5 i) + 5 x^{2} i x + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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 $1$ .

$y = x^{3} x + x^{3} (- 5 i) + 5 x^{2} i x + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{3 + 1} + x^{3} (- 5 i) + 5 x^{2} i x + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{2} i x + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 (x \cdot x^{2}) i + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

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

$y = x^{4} + x^{3} (- 5 i) + 5 (x \cdot x^{2}) i + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{1 + 2} i + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Add $1$ and $2$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 5 x^{2} i (- 5 i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $5 x^{2} i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} i i + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} (i i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} (i i) + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} i^{1 + 1} + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} i^{2} + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i - 25 x^{2} \cdot - 1 + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $- 1$ by $- 25$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{2} x + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 (x \cdot x^{2}) + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 (x \cdot x^{2}) + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{1 + 2} + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Add $1$ and $2$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} + 10 x^{2} (- 5 i) + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $- 5$ by $10$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x i x + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 (x \cdot x) i + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $x$ by $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 50 x i (- 5 i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $50 x i (- 5 i)$ .

Multiply $- 5$ by $50$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x i i + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x (i i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x (i i) + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x i^{1 + 1} + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x i^{2} + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i - 250 x \cdot - 1 + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $- 1$ by $- 250$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x \cdot x + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

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

Move $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 (x \cdot x) + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $x$ by $x$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} + 29 x (- 5 i) + 145 i x + 145 i (- 5 i)$

Multiply $- 5$ by $29$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 145 i (- 5 i)$

Multiply $145 i (- 5 i)$ .

Multiply $- 5$ by $145$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 i i$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 (i i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 (i i)$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 i^{2}$

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

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x - 725 \cdot - 1$

Multiply $- 725$ by $- 1$ .

$y = x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + x^{3} (- 5 i) + 5 x^{3} i + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$ .

Reorder the factors in the terms $x^{3} (- 5 i)$ and $5 x^{3} i$ .

$y = x^{4} - 5 i x^{3} + 5 i x^{3} + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

Add $- 5 i x^{3}$ and $5 i x^{3}$ .

$y = x^{4} + 0 + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

Add $x^{4}$ and $0$ .

$y = x^{4} + 25 x^{2} + 10 x^{3} - 50 x^{2} i + 50 x^{2} i + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

Add $- 50 x^{2} i$ and $50 x^{2} i$ .

$y = x^{4} + 25 x^{2} + 10 x^{3} + 0 + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

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

$y = x^{4} + 25 x^{2} + 10 x^{3} + 250 x + 29 x^{2} - 145 x i + 145 i x + 725$

Reorder the factors in the terms $- 145 x i$ and $145 i x$ .

$y = x^{4} + 25 x^{2} + 10 x^{3} + 250 x + 29 x^{2} - 145 i x + 145 i x + 725$

Add $- 145 i x$ and $145 i x$ .

$y = x^{4} + 25 x^{2} + 10 x^{3} + 250 x + 29 x^{2} + 0 + 725$

Add $x^{4} + 25 x^{2} + 10 x^{3} + 250 x + 29 x^{2}$ and $0$ .

$y = x^{4} + 25 x^{2} + 10 x^{3} + 250 x + 29 x^{2} + 725$

Add $25 x^{2}$ and $29 x^{2}$ .

$y = x^{4} + 54 x^{2} + 10 x^{3} + 250 x + 725$

Move $54 x^{2}$ .

$y = x^{4} + 10 x^{3} + 54 x^{2} + 250 x + 725$

Step 8

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

Related Questions

Support LCMGCF.com by disabling your adblocker.