Find the Equation Given the Roots 2- square root of 6 , 2+ square root of 6 , 7-i

$2 - \sqrt{6}$ , $2 + \sqrt{6}$ , $7 - i$

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 $x = 2 - \sqrt{6}$ was found by solving for $x$ when $x - (2 - \sqrt{6}) = y$ and $y = 0$ .

The factor is $x - 2 + \sqrt{6}$

Step 3

The root at $x = 2 + \sqrt{6}$ was found by solving for $x$ when $x - (2 + \sqrt{6}) = y$ and $y = 0$ .

The factor is $x - 2 - \sqrt{6}$

Step 4

The root at $x = 7 - i$ was found by solving for $x$ when $x - (7 - i) = y$ and $y = 0$ .

The factor is $x - 7 + i$

Step 5

The root at $x = 7 + i$ was found by solving for $x$ when $x - (7 + i) = y$ and $y = 0$ .

The factor is $x - 7 - i$

Step 6

Combine all the factors into a single equation.

$y = (x - 2 + \sqrt{6}) (x - 2 - \sqrt{6}) (x - 7 + i) (x - 7 - i)$

Step 7

Multiply all the factors to simplify the equation $y = (x - 2 + \sqrt{6}) (x - 2 - \sqrt{6}) (x - 7 + i) (x - 7 - i)$ .

Expand $(x - 2 + \sqrt{6}) (x - 2 - \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 2 + x (- \sqrt{6}) - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} x + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 2 + x (- \sqrt{6}) - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} x + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})$ .

Reorder the factors in the terms $x (- \sqrt{6})$ and $\sqrt{6} x$ .

$y = (x \cdot x + x \cdot - 2 - x \sqrt{6} - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + x \sqrt{6} + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Add $- x \sqrt{6}$ and $x \sqrt{6}$ .

$y = (x \cdot x + x \cdot - 2 + 0 - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Add $x \cdot x + x \cdot - 2$ and $0$ .

$y = (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

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

$y = (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 2 (- \sqrt{6}) + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Multiply $- 2$ by $- 2$ .

$y = (x^{2} - 2 x - 2 x + 4 - 2 (- \sqrt{6}) + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Multiply $- 1$ by $- 2$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} + \sqrt{6} \cdot - 2 + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Move $- 2$ to the left of $\sqrt{6}$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \cdot \sqrt{6} + \sqrt{6} (- \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Multiply $\sqrt{6} (- \sqrt{6})$ .

Raise $\sqrt{6}$ to the power of $1$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - (\sqrt{6} \sqrt{6})) (x - 7 + i) (x - 7 - i)$

Raise $\sqrt{6}$ to the power of $1$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - (\sqrt{6} \sqrt{6})) (x - 7 + i) (x - 7 - i)$

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

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - {\sqrt{6}}^{1 + 1}) (x - 7 + i) (x - 7 - i)$

Add $1$ and $1$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - {\sqrt{6}}^{2}) (x - 7 + i) (x - 7 - i)$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - {(6^{\frac{1}{2}})}^{2}) (x - 7 + i) (x - 7 - i)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6^{\frac{1}{2} \cdot 2}) (x - 7 + i) (x - 7 - i)$

Combine $\frac{1}{2}$ and $2$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6^{\frac{2}{2}}) (x - 7 + i) (x - 7 - i)$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6^{\frac{2}{2}}) (x - 7 + i) (x - 7 - i)$

Rewrite the expression.

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6) (x - 7 + i) (x - 7 - i)$

Evaluate the exponent.

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 1 \cdot 6) (x - 7 + i) (x - 7 - i)$

Multiply $- 1$ by $6$ .

$y = (x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6) (x - 7 + i) (x - 7 - i)$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 2 x - 2 x + 4 + 2 \sqrt{6} - 2 \sqrt{6} - 6$ .

Subtract $2 \sqrt{6}$ from $2 \sqrt{6}$ .

$y = (x^{2} - 2 x - 2 x + 4 + 0 - 6) (x - 7 + i) (x - 7 - i)$

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

$y = (x^{2} - 2 x - 2 x + 4 - 6) (x - 7 + i) (x - 7 - i)$

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

$y = (x^{2} - 4 x + 4 - 6) (x - 7 + i) (x - 7 - i)$

Subtract $6$ from $4$ .

$y = (x^{2} - 4 x - 2) (x - 7 + i) (x - 7 - i)$

Expand $(x^{2} - 4 x - 2) (x - 7 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 7 + x^{2} i - 4 x \cdot x - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

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 $1$ .

$y = (x^{2} x + x^{2} \cdot - 7 + x^{2} i - 4 x \cdot x - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

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

$y = (x^{2 + 1} + x^{2} \cdot - 7 + x^{2} i - 4 x \cdot x - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 7 + x^{2} i - 4 x \cdot x - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

Move $- 7$ to the left of $x^{2}$ .

$y = (x^{3} - 7 \cdot x^{2} + x^{2} i - 4 x \cdot x - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

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

Move $x$ .

$y = (x^{3} - 7 x^{2} + x^{2} i - 4 (x \cdot x) - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

Multiply $x$ by $x$ .

$y = (x^{3} - 7 x^{2} + x^{2} i - 4 x^{2} - 4 x \cdot - 7 - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

Multiply $- 7$ by $- 4$ .

$y = (x^{3} - 7 x^{2} + x^{2} i - 4 x^{2} + 28 x - 4 x i - 2 x - 2 \cdot - 7 - 2 i) (x - 7 - i)$

Multiply $- 2$ by $- 7$ .

$y = (x^{3} - 7 x^{2} + x^{2} i - 4 x^{2} + 28 x - 4 x i - 2 x + 14 - 2 i) (x - 7 - i)$

Simplify by adding terms.

Subtract $4 x^{2}$ from $- 7 x^{2}$ .

$y = (x^{3} - 11 x^{2} + x^{2} i + 28 x - 4 x i - 2 x + 14 - 2 i) (x - 7 - i)$

Subtract $2 x$ from $28 x$ .

$y = (x^{3} - 11 x^{2} + x^{2} i - 4 x i + 26 x + 14 - 2 i) (x - 7 - i)$

Expand $(x^{3} - 11 x^{2} + x^{2} i - 4 x i + 26 x + 14 - 2 i) (x - 7 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 7 + x^{3} (- i) - 11 x^{2} x - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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 $1$ .

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

$y = x^{3 + 1} + x^{3} \cdot - 7 + x^{3} (- i) - 11 x^{2} x - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 7 + x^{3} (- i) - 11 x^{2} x - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Move $- 7$ to the left of $x^{3}$ .

$y = x^{4} - 7 \cdot x^{3} + x^{3} (- i) - 11 x^{2} x - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

Move $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 (x \cdot x^{2}) - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

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

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{1 + 2} - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Add $1$ and $2$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} - 11 x^{2} \cdot - 7 - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 7$ by $- 11$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} - 11 x^{2} (- i) + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 1$ by $- 11$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{2} i x + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

Move $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x \cdot x^{2} i + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

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

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{1 + 2} i + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Add $1$ and $2$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i + x^{2} i \cdot - 7 + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Move $- 7$ to the left of $x^{2} i$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 \cdot (x^{2} i) + x^{2} i (- i) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} (- (i i)) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} (- i^{1 + 1}) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Add $1$ and $1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} (- i^{2}) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} (1) - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} \cdot 1 - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $x^{2}$ by $1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x i x - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

Move $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 (x \cdot x) i - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i - 4 x i \cdot - 7 - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 7$ by $- 4$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x i (- i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 4 x i (- i)$ .

Multiply $- 1$ by $- 4$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i + 4 x i i + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i + 4 x (i i) + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i + 4 x i^{1 + 1} + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Add $1$ and $1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i + 4 x i^{2} + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i + 4 x \cdot - 1 + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 1$ by $4$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x \cdot x + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

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

Move $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 (x \cdot x) + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $x$ by $x$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} + 26 x \cdot - 7 + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 7$ by $26$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x + 26 x (- i) + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 1$ by $26$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x + 14 \cdot - 7 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $14$ by $- 7$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 + 14 (- i) - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 1$ by $14$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x - 2 i \cdot - 7 - 2 i (- i)$

Multiply $- 7$ by $- 2$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2 i (- i)$

Multiply $- 2 i (- i)$ .

Multiply $- 1$ by $- 2$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 i i$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 (i i)$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 (i i)$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 i^{2}$

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

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$y = x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2$

Simplify by adding terms.

Combine the opposite terms in $x^{4} - 7 x^{3} + x^{3} (- i) - 11 x^{3} + 77 x^{2} + 11 x^{2} i + x^{3} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2$ .

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

$y = x^{4} - 7 x^{3} - i x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i + i x^{3} - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2$

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

$y = x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i + 0 - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2$

Add $x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i$ and $0$ .

$y = x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 14 i - 2 i x + 14 i - 2$

Add $- 14 i$ and $14 i$ .

$y = x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 2 i x + 0 - 2$

Add $x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 2 i x$ and $0$ .

$y = x^{4} - 7 x^{3} - 11 x^{3} + 77 x^{2} + 11 x^{2} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Subtract $11 x^{3}$ from $- 7 x^{3}$ .

$y = x^{4} - 18 x^{3} + 77 x^{2} + 11 x^{2} i - 7 x^{2} i + x^{2} - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Add $77 x^{2}$ and $x^{2}$ .

$y = x^{4} - 18 x^{3} + 78 x^{2} + 11 x^{2} i - 7 x^{2} i - 4 x^{2} i + 28 x i - 4 x + 26 x^{2} - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Add $78 x^{2}$ and $26 x^{2}$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} + 11 x^{2} i - 7 x^{2} i - 4 x^{2} i + 28 x i - 4 x - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Subtract $7 x^{2} i$ from $11 x^{2} i$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} + 4 x^{2} i - 4 x^{2} i + 28 x i - 4 x - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Combine the opposite terms in $x^{4} - 18 x^{3} + 104 x^{2} + 4 x^{2} i - 4 x^{2} i + 28 x i - 4 x - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$ .

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

$y = x^{4} - 18 x^{3} + 104 x^{2} + 0 + 28 x i - 4 x - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Add $x^{4} - 18 x^{3} + 104 x^{2}$ and $0$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} + 28 x i - 4 x - 182 x - 26 x i + 14 x - 98 - 2 i x - 2$

Subtract $26 x i$ from $28 x i$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} + 2 x i - 4 x - 182 x + 14 x - 98 - 2 i x - 2$

Combine the opposite terms in $x^{4} - 18 x^{3} + 104 x^{2} + 2 x i - 4 x - 182 x + 14 x - 98 - 2 i x - 2$ .

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

$y = x^{4} - 18 x^{3} + 104 x^{2} + 2 i x - 4 x - 182 x + 14 x - 98 - 2 i x - 2$

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

$y = x^{4} - 18 x^{3} + 104 x^{2} - 4 x - 182 x + 14 x - 98 + 0 - 2$

Add $x^{4} - 18 x^{3} + 104 x^{2} - 4 x - 182 x + 14 x - 98$ and $0$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} - 4 x - 182 x + 14 x - 98 - 2$

Subtract $182 x$ from $- 4 x$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} - 186 x + 14 x - 98 - 2$

Add $- 186 x$ and $14 x$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} - 172 x - 98 - 2$

Subtract $2$ from $- 98$ .

$y = x^{4} - 18 x^{3} + 104 x^{2} - 172 x - 100$

Step 8

Find the Equation Given the Roots 2- square root of 6 , 2+ square root of 6 , 7-i

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