Find the Equation Given the Roots -5 , 4+3i

$- 5$ , $4 + 3 i$

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

The factor is $x + 5$

Step 3

The root at $x = 4 + 3 i$ was found by solving for $x$ when $x - (4 + 3 i) = y$ and $y = 0$ .

The factor is $x - 4 - 3 i$

Step 4

The root at $x = 4 - 3 i$ was found by solving for $x$ when $x - (4 - 3 i) = y$ and $y = 0$ .

The factor is $x - 4 + 3 i$

Step 5

Combine all the factors into a single equation.

$y = (x + 5) (x - 4 - 3 i) (x - 4 + 3 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 5) (x - 4 - 3 i) (x - 4 + 3 i)$ .

Expand $(x + 5) (x - 4 - 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 4 + x (- 3 i) + 5 x + 5 \cdot - 4 + 5 (- 3 i)) (x - 4 + 3 i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $5$ by $- 4$ .

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

Multiply $- 3$ by $5$ .

$y = (x^{2} - 4 x + x (- 3 i) + 5 x - 20 - 15 i) (x - 4 + 3 i)$

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

$y = (x^{2} + x (- 3 i) + x - 20 - 15 i) (x - 4 + 3 i)$

Expand $(x^{2} + x (- 3 i) + x - 20 - 15 i) (x - 4 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 4 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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 $1$ .

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

$y = x^{2 + 1} + x^{2} \cdot - 4 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 4 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Move $- 4$ to the left of $x^{2}$ .

$y = x^{3} - 4 \cdot x^{2} + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

$y = x^{3} - 4 x^{2} + 3 \cdot (x^{2} i) + x (- 3 i) x + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

Move $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x \cdot x (- 3 i) + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + x (- 3 i) \cdot - 4 + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $- 4$ by $- 3$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + x (12 i) + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Move $12$ to the left of $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 \cdot (x i) + x (- 3 i) (3 i) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $x (- 3 i) (3 i)$ .

Multiply $3$ by $- 3$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 i) i + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 (i i)) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 (i i)) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 i^{1 + 1}) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Add $1$ and $1$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 i^{2}) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x (- 9 \cdot - 1) + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $- 9$ by $- 1$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + x \cdot 9 + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Move $9$ to the left of $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 \cdot x + x \cdot x + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} + x \cdot - 4 + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Move $- 4$ to the left of $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 \cdot x + x (3 i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Move $3$ to the left of $x$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 \cdot (x i) - 20 x - 20 \cdot - 4 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $- 20$ by $- 4$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 20 (3 i) - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $3$ by $- 20$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x - 15 i \cdot - 4 - 15 i (3 i)$

Multiply $- 4$ by $- 15$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 15 i (3 i)$

Multiply $- 15 i (3 i)$ .

Multiply $3$ by $- 15$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 i i$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 (i i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 (i i)$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 i^{2}$

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

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i - 45 \cdot - 1$

Multiply $- 45$ by $- 1$ .

$y = x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i + 45$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 4 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i + 45$ .

Reorder the factors in the terms $3 x^{2} i$ and $x^{2} (- 3 i)$ .

$y = x^{3} - 4 x^{2} + 3 i x^{2} - 3 i x^{2} + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i + 45$

Subtract $3 i x^{2}$ from $3 i x^{2}$ .

$y = x^{3} - 4 x^{2} + 0 + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i + 45$

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

$y = x^{3} - 4 x^{2} + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 60 i - 15 i x + 60 i + 45$

Add $- 60 i$ and $60 i$ .

$y = x^{3} - 4 x^{2} + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 15 i x + 0 + 45$

Add $x^{3} - 4 x^{2} + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 15 i x$ and $0$ .

$y = x^{3} - 4 x^{2} + 12 x i + 9 x + x^{2} - 4 x + 3 x i - 20 x + 80 - 15 i x + 45$

Add $- 4 x^{2}$ and $x^{2}$ .

$y = x^{3} - 3 x^{2} + 12 x i + 9 x - 4 x + 3 x i - 20 x + 80 - 15 i x + 45$

Add $12 x i$ and $3 x i$ .

$y = x^{3} - 3 x^{2} + 15 x i + 9 x - 4 x - 20 x + 80 - 15 i x + 45$

Combine the opposite terms in $x^{3} - 3 x^{2} + 15 x i + 9 x - 4 x - 20 x + 80 - 15 i x + 45$ .

Reorder the factors in the terms $15 x i$ and $- 15 i x$ .

$y = x^{3} - 3 x^{2} + 15 i x + 9 x - 4 x - 20 x + 80 - 15 i x + 45$

Subtract $15 i x$ from $15 i x$ .

$y = x^{3} - 3 x^{2} + 9 x - 4 x - 20 x + 80 + 0 + 45$

Add $x^{3} - 3 x^{2} + 9 x - 4 x - 20 x + 80$ and $0$ .

$y = x^{3} - 3 x^{2} + 9 x - 4 x - 20 x + 80 + 45$

Subtract $4 x$ from $9 x$ .

$y = x^{3} - 3 x^{2} + 5 x - 20 x + 80 + 45$

Subtract $20 x$ from $5 x$ .

$y = x^{3} - 3 x^{2} - 15 x + 80 + 45$

Add $80$ and $45$ .

$y = x^{3} - 3 x^{2} - 15 x + 125$

Step 7

Find the Equation Given the Roots -5 , 4+3i

Related Questions

Support LCMGCF.com by disabling your adblocker.