Find the Equation Given the Roots 1+5i , 4+ square root of 3

$1 + 5 i$ , $4 + \sqrt{3}$

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 $x = 1 + 5 i$ was found by solving for $x$ when $x - (1 + 5 i) = y$ and $y = 0$ .

The factor is $x - 1 - 5 i$

Step 3

The root at $x = 1 - 5 i$ was found by solving for $x$ when $x - (1 - 5 i) = y$ and $y = 0$ .

The factor is $x - 1 + 5 i$

Step 4

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

The factor is $x - 4 - \sqrt{3}$

Step 5

Combine all the factors into a single equation.

$y = (x - 1 - 5 i) (x - 1 + 5 i) (x - 4 - \sqrt{3})$

Step 6

Multiply all the factors to simplify the equation $y = (x - 1 - 5 i) (x - 1 + 5 i) (x - 4 - \sqrt{3})$ .

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

$y = (x \cdot x + x \cdot - 1 + x (5 i) - 1 x - 1 \cdot - 1 - 1 (5 i) - 5 i x - 5 i \cdot - 1 - 5 i (5 i)) (x - 4 - \sqrt{3})$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (5 i) - 1 x - 1 \cdot - 1 - 1 (5 i) - 5 i x - 5 i \cdot - 1 - 5 i (5 i)$ .

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

$y = (x \cdot x + x \cdot - 1 + 5 i x - 1 x - 1 \cdot - 1 - 1 (5 i) - 5 i x - 5 i \cdot - 1 - 5 i (5 i)) (x - 4 - \sqrt{3})$

Subtract $5 i x$ from $5 i x$ .

$y = (x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 (5 i) - 5 i \cdot - 1 - 5 i (5 i)) (x - 4 - \sqrt{3})$

Add $x \cdot x + x \cdot - 1$ and $0$ .

$y = (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (5 i) - 5 i \cdot - 1 - 5 i (5 i)) (x - 4 - \sqrt{3})$

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $5$ by $- 1$ .

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

Multiply $- 1$ by $- 5$ .

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

Multiply $- 5 i (5 i)$ .

Multiply $5$ by $- 5$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 25$ by $- 1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - x + 1 - 5 i + 5 i + 25$ .

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

$y = (x^{2} - x - x + 1 + 0 + 25) (x - 4 - \sqrt{3})$

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

$y = (x^{2} - x - x + 1 + 25) (x - 4 - \sqrt{3})$

Subtract $x$ from $- x$ .

$y = (x^{2} - 2 x + 1 + 25) (x - 4 - \sqrt{3})$

Add $1$ and $25$ .

$y = (x^{2} - 2 x + 26) (x - 4 - \sqrt{3})$

Expand $(x^{2} - 2 x + 26) (x - 4 - \sqrt{3})$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 4 + x^{2} (- \sqrt{3}) - 2 x \cdot x - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 4 + x^{2} (- \sqrt{3}) - 2 x \cdot x - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

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

$y = x^{2 + 1} + x^{2} \cdot - 4 + x^{2} (- \sqrt{3}) - 2 x \cdot x - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 4 + x^{2} (- \sqrt{3}) - 2 x \cdot x - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

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

$y = x^{3} - 4 \cdot x^{2} + x^{2} (- \sqrt{3}) - 2 x \cdot x - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

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

Move $x$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 (x \cdot x) - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

Multiply $x$ by $x$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 x^{2} - 2 x \cdot - 4 - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

Multiply $- 4$ by $- 2$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 x^{2} + 8 x - 2 x (- \sqrt{3}) + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

Multiply $- 1$ by $- 2$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 x^{2} + 8 x + 2 x \sqrt{3} + 26 x + 26 \cdot - 4 + 26 (- \sqrt{3})$

Multiply $26$ by $- 4$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 x^{2} + 8 x + 2 x \sqrt{3} + 26 x - 104 + 26 (- \sqrt{3})$

Multiply $- 1$ by $26$ .

$y = x^{3} - 4 x^{2} + x^{2} (- \sqrt{3}) - 2 x^{2} + 8 x + 2 x \sqrt{3} + 26 x - 104 - 26 \sqrt{3}$

Simplify by adding terms.

Subtract $2 x^{2}$ from $- 4 x^{2}$ .

$y = x^{3} - 6 x^{2} + x^{2} (- \sqrt{3}) + 8 x + 2 x \sqrt{3} + 26 x - 104 - 26 \sqrt{3}$

Add $8 x$ and $26 x$ .

$y = x^{3} - 6 x^{2} + x^{2} (- \sqrt{3}) + 2 x \sqrt{3} + 34 x - 104 - 26 \sqrt{3}$

Reorder factors in $x^{3} - 6 x^{2} + x^{2} (- \sqrt{3}) + 2 x \sqrt{3} + 34 x - 104 - 26 \sqrt{3}$ .

$y = x^{3} - 6 x^{2} - x^{2} \sqrt{3} + 2 x \sqrt{3} + 34 x - 104 - 26 \sqrt{3}$

Step 7

Find the Equation Given the Roots 1+5i , 4+ square root of 3

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