Find the Equation Given the Roots -1/3 , square root of 11 , -2i

$- \frac{1}{3}$ , $\sqrt{11}$ , $- 2 i$

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 $x = - \frac{1}{3}$ was found by solving for $x$ when $x - (- \frac{1}{3}) = y$ and $y = 0$ .

The factor is $x + \frac{1}{3}$

Step 3

The root at $x = \sqrt{11}$ was found by solving for $x$ when $x - (\sqrt{11}) = y$ and $y = 0$ .

The factor is $x - \sqrt{11}$

Step 4

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

The factor is $x + 2 i$

Step 5

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

The factor is $x - 2 i$

Step 6

Combine all the factors into a single equation.

$y = (x + \frac{1}{3}) (x - \sqrt{11}) (x + 2 i) (x - 2 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x + \frac{1}{3}) (x - \sqrt{11}) (x + 2 i) (x - 2 i)$ .

Expand $(x + \frac{1}{3}) (x - \sqrt{11})$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - \sqrt{11}) + \frac{1}{3} \cdot (x - \sqrt{11})) (x + 2 i) (x - 2 i)$

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{11}) + \frac{1}{3} \cdot (x - \sqrt{11})) (x + 2 i) (x - 2 i)$

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{11}) + \frac{1}{3} x + \frac{1}{3} \cdot (- \sqrt{11})) (x + 2 i) (x - 2 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x (- \sqrt{11}) + \frac{1}{3} x + \frac{1}{3} \cdot (- \sqrt{11})) (x + 2 i) (x - 2 i)$

Combine $\frac{1}{3}$ and $x$ .

$y = (x^{2} + x (- \sqrt{11}) + \frac{x}{3} + \frac{1}{3} \cdot (- \sqrt{11})) (x + 2 i) (x - 2 i)$

Combine $\frac{1}{3}$ and $\sqrt{11}$ .

$y = (x^{2} + x (- \sqrt{11}) + \frac{x}{3} - \frac{\sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Subtract $\frac{\sqrt{11}}{3}$ from $x (- \sqrt{11})$ .

Reorder $x$ and $- 1$ .

$y = (x^{2} + \frac{x}{3} - 1 \cdot (x \sqrt{11}) - \frac{\sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

To write $- 1 \cdot x \sqrt{11}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = (x^{2} + \frac{x}{3} - 1 \cdot (x \sqrt{11}) \cdot \frac{3}{3} - \frac{\sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Combine $- 1 \cdot x \sqrt{11}$ and $\frac{3}{3}$ .

$y = (x^{2} + \frac{x}{3} + \frac{- 1 \cdot (x \sqrt{11}) \cdot 3}{3} - \frac{\sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Combine the numerators over the common denominator.

$y = (x^{2} + \frac{x}{3} + \frac{- 1 \cdot (x \sqrt{11}) \cdot 3 - \sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = (\frac{x}{3} + x^{2} \cdot \frac{3}{3} + \frac{- 1 \cdot (x \sqrt{11}) \cdot 3 - \sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Combine $x^{2}$ and $\frac{3}{3}$ .

$y = (\frac{x}{3} + \frac{x^{2} \cdot 3}{3} + \frac{- 1 \cdot (x \sqrt{11}) \cdot 3 - \sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Combine the numerators over the common denominator.

$y = (\frac{x}{3} + \frac{x^{2} \cdot 3 - 1 \cdot (x \sqrt{11}) \cdot 3 - \sqrt{11}}{3}) (x + 2 i) (x - 2 i)$

Combine the numerators over the common denominator.

$y = \frac{x + x^{2} \cdot 3 - 1 \cdot (x \sqrt{11}) \cdot 3 - \sqrt{11}}{3} \cdot ((x + 2 i) (x - 2 i))$

Simplify the numerator.

Reorder terms.

$y = \frac{3 x^{2} - 1 \cdot (3 x \sqrt{11}) + x - \sqrt{11}}{3} \cdot ((x + 2 i) (x - 2 i))$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(3 x^{2} - 1 \cdot (3 x \sqrt{11})) + x - \sqrt{11}}{3} \cdot ((x + 2 i) (x - 2 i))$

Factor out the greatest common factor (GCF) from each group.

$y = \frac{3 x (x - \sqrt{11}) + 1 (x - \sqrt{11})}{3} \cdot ((x + 2 i) (x - 2 i))$

Factor the polynomial by factoring out the greatest common factor, $x - \sqrt{11}$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1)}{3} \cdot ((x + 2 i) (x - 2 i))$

Simplify terms.

Apply the distributive property.

$y = (\frac{(x - \sqrt{11}) (3 x + 1)}{3} \cdot x + \frac{(x - \sqrt{11}) (3 x + 1)}{3} \cdot (2 i)) (x - 2 i)$

Combine $\frac{(x - \sqrt{11}) (3 x + 1)}{3}$ and $x$ .

$y = (\frac{(x - \sqrt{11}) (3 x + 1) x}{3} + \frac{(x - \sqrt{11}) (3 x + 1)}{3} \cdot (2 i)) (x - 2 i)$

Multiply $\frac{(x - \sqrt{11}) (3 x + 1)}{3} (2 i)$ .

Combine $2$ and $\frac{(x - \sqrt{11}) (3 x + 1)}{3}$ .

$y = (\frac{(x - \sqrt{11}) (3 x + 1) x}{3} + \frac{2 ((x - \sqrt{11}) (3 x + 1))}{3} \cdot i) (x - 2 i)$

Combine $\frac{2 ((x - \sqrt{11}) (3 x + 1))}{3}$ and $i$ .

$y = (\frac{(x - \sqrt{11}) (3 x + 1) x}{3} + \frac{2 ((x - \sqrt{11}) (3 x + 1)) i}{3}) (x - 2 i)$

Simplify terms.

Combine the numerators over the common denominator.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + 2 (x - \sqrt{11}) (3 x + 1) i}{3} \cdot (x - 2 i)$

Simplify each term.

Apply the distributive property.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 x + 2 (- \sqrt{11})) (3 x + 1) i}{3} \cdot (x - 2 i)$

Multiply $- 1$ by $2$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 x - 2 \sqrt{11}) (3 x + 1) i}{3} \cdot (x - 2 i)$

Expand $(2 x - 2 \sqrt{11}) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 x (3 x + 1) - 2 \sqrt{11} (3 x + 1)) i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 x (3 x) + 2 x \cdot 1 - 2 \sqrt{11} (3 x + 1)) i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 x (3 x) + 2 x \cdot 1 - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 \cdot (3 x \cdot x) + 2 x \cdot 1 - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

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

Move $x$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 \cdot (3 (x \cdot x)) + 2 x \cdot 1 - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Multiply $x$ by $x$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (2 \cdot (3 x^{2}) + 2 x \cdot 1 - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Multiply $2$ by $3$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (6 x^{2} + 2 x \cdot 1 - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Multiply $2$ by $1$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (6 x^{2} + 2 x - 2 \sqrt{11} (3 x) - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Multiply $3$ by $- 2$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (6 x^{2} + 2 x - 6 \sqrt{11} x - 2 \sqrt{11} \cdot 1) i}{3} \cdot (x - 2 i)$

Multiply $- 2$ by $1$ .

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + (6 x^{2} + 2 x - 6 \sqrt{11} x - 2 \sqrt{11}) i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{(x - \sqrt{11}) (3 x + 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Simplify the numerator.

Expand $(x - \sqrt{11}) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{(x (3 x + 1) - \sqrt{11} (3 x + 1)) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{(x (3 x) + x \cdot 1 - \sqrt{11} (3 x + 1)) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{(x (3 x) + x \cdot 1 - \sqrt{11} (3 x) - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{(3 x \cdot x + x \cdot 1 - \sqrt{11} (3 x) - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

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

Move $x$ .

$y = \frac{(3 (x \cdot x) + x \cdot 1 - \sqrt{11} (3 x) - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $x$ by $x$ .

$y = \frac{(3 x^{2} + x \cdot 1 - \sqrt{11} (3 x) - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $x$ by $1$ .

$y = \frac{(3 x^{2} + x - \sqrt{11} (3 x) - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $3$ by $- 1$ .

$y = \frac{(3 x^{2} + x - 3 \sqrt{11} x - \sqrt{11} \cdot 1) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $- 1$ by $1$ .

$y = \frac{(3 x^{2} + x - 3 \sqrt{11} x - \sqrt{11}) x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Apply the distributive property.

$y = \frac{3 x^{2} x + x \cdot x - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Simplify.

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

Move $x$ .

$y = \frac{3 (x \cdot x^{2}) + x \cdot x - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

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

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

$y = \frac{3 (x \cdot x^{2}) + x \cdot x - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

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

$y = \frac{3 x^{1 + 2} + x \cdot x - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Add $1$ and $2$ .

$y = \frac{3 x^{3} + x \cdot x - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $x$ by $x$ .

$y = \frac{3 x^{3} + x^{2} - 3 \sqrt{11} x \cdot x - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

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

Move $x$ .

$y = \frac{3 x^{3} + x^{2} - 3 \sqrt{11} (x \cdot x) - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Multiply $x$ by $x$ .

$y = \frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (x - 2 i)$

Simplify terms.

Apply the distributive property.

$y = \frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot x + \frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (- 2 i)$

Combine $\frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3}$ and $x$ .

$y = \frac{(3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) x}{3} + \frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} \cdot (- 2 i)$

Multiply $\frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3} (- 2 i)$ .

Combine $- 2$ and $\frac{3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i}{3}$ .

$y = \frac{(3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) x}{3} + \frac{- 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i)}{3} \cdot i$

Combine $\frac{- 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i)}{3}$ and $i$ .

Combine the numerators over the common denominator.

$y = \frac{(3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Simplify each term.

Apply the distributive property.

$y = \frac{3 x^{3} x + x^{2} x - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Simplify.

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$y = \frac{3 (x \cdot x^{3}) + x^{2} x - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Multiply $x$ by $x^{3}$ .

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

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

$y = \frac{3 x^{1 + 3} + x^{2} x - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Add $1$ and $3$ .

$y = \frac{3 x^{4} + x^{2} x - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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.

$y = \frac{3 x^{4} + x^{2 + 1} - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Add $2$ and $1$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{2} x - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} (x \cdot x^{2}) - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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.

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{1 + 2} - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Add $1$ and $2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x \cdot x + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} (x \cdot x) + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{2} i x + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 (x \cdot x^{2}) i + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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.

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{1 + 2} i + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Add $1$ and $2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x i x - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 (x \cdot x) i - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x i x - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} (x \cdot x) i - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 2 (3 x^{3} + x^{2} - 3 \sqrt{11} x^{2} - \sqrt{11} x + 6 x^{2} i + 2 x i - 6 \sqrt{11} x i - 2 \sqrt{11} i) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 2 (3 x^{3}) - 2 x^{2} - 2 (- 3 \sqrt{11} x^{2}) - 2 (- \sqrt{11} x) - 2 (6 x^{2} i) - 2 (2 x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Simplify.

Multiply $3$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} - 2 (- 3 \sqrt{11} x^{2}) - 2 (- \sqrt{11} x) - 2 (6 x^{2} i) - 2 (2 x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $- 3$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) - 2 (- \sqrt{11} x) - 2 (6 x^{2} i) - 2 (2 x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $- 1$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) + 2 (\sqrt{11} x) - 2 (6 x^{2} i) - 2 (2 x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $6$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) + 2 (\sqrt{11} x) - 12 (x^{2} i) - 2 (2 x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $2$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) + 2 (\sqrt{11} x) - 12 (x^{2} i) - 4 (x i) - 2 (- 6 \sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $- 6$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) + 2 (\sqrt{11} x) - 12 (x^{2} i) - 4 (x i) + 12 (\sqrt{11} x i) - 2 (- 2 \sqrt{11} i)) i}{3}$

Multiply $- 2$ by $- 2$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 (\sqrt{11} x^{2}) + 2 (\sqrt{11} x) - 12 (x^{2} i) - 4 (x i) + 12 (\sqrt{11} x i) + 4 (\sqrt{11} i)) i}{3}$

Remove parentheses.

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + (- 6 x^{3} - 2 x^{2} + 6 \sqrt{11} x^{2} + 2 \sqrt{11} x - 12 x^{2} i - 4 x i + 12 \sqrt{11} x i + 4 \sqrt{11} i) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i i - 4 x i i + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

Simplify.

Multiply $- 12 x^{2} i i$ .

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} (i i) - 4 x i i + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} (i i) - 4 x i i + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{1 + 1} - 4 x i i + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i i + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

Multiply $- 4 x i i$ .

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x (i i) + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x (i i) + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{1 + 1} + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i i + 4 \sqrt{11} i i}{3}$

Multiply $12 \sqrt{11} x i i$ .

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x (i i) + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x (i i) + 4 \sqrt{11} i i}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{1 + 1} + 4 \sqrt{11} i i}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i i}{3}$

Multiply $4 \sqrt{11} i i$ .

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} (i i)}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} (i i)}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{1 + 1}}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} i^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{2}}{3}$

Simplify each term.

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i - 12 x^{2} \cdot - 1 - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{2}}{3}$

Multiply $- 1$ by $- 12$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} - 4 x i^{2} + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{2}}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} - 4 x \cdot - 1 + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{2}}{3}$

Multiply $- 1$ by $- 4$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x + 12 \sqrt{11} x i^{2} + 4 \sqrt{11} i^{2}}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x + 12 \sqrt{11} x \cdot - 1 + 4 \sqrt{11} i^{2}}{3}$

Multiply $- 1$ by $12$ .

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x + 4 \sqrt{11} \cdot - 1}{3}$

Multiply $- 1$ by $4$ .

Simplify by adding terms.

Combine the opposite terms in $3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 6 x^{3} i + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 6 x^{3} i - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}$ .

Subtract $6 x^{3} i$ from $6 x^{3} i$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + 0 - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Add $3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x$ and $0$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 2 x^{2} i - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x - 2 x^{2} i + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + 0 + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Add $3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x$ and $0$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 6 \sqrt{11} x^{2} i - 2 \sqrt{11} i x + 6 \sqrt{11} x^{2} i + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Add $- 6 \sqrt{11} x^{2} i$ and $6 \sqrt{11} x^{2} i$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 2 \sqrt{11} i x + 0 + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

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

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 2 \sqrt{11} i x + 2 \sqrt{11} x i + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Reorder the factors in the terms $- 2 \sqrt{11} i x$ and $2 \sqrt{11} x i$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} - 2 i x \sqrt{11} + 2 i x \sqrt{11} + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Add $- 2 i x \sqrt{11}$ and $2 i x \sqrt{11}$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 0 + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Add $3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2}$ and $0$ .

$y = \frac{3 x^{4} + x^{3} - 3 \sqrt{11} x^{3} - \sqrt{11} x^{2} + 12 x^{2} + 4 x - 12 \sqrt{11} x - 4 \sqrt{11}}{3}$

Reorder terms.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11} + 4 x - 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11} + 4 x - 4 \sqrt{11}}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11} + 4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11} + 4 x}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11} - 12 x \sqrt{11}}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11} - x^{2} \sqrt{11}}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2} - 3 x^{3} \sqrt{11}}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3} + 12 x^{2}}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3} + 12 x^{2}}{3}$ into two fractions.

$y = \frac{3 x^{4} + x^{3}}{3} + \frac{12 x^{2}}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Split the fraction $\frac{3 x^{4} + x^{3}}{3}$ into two fractions.

$y = \frac{3 x^{4}}{3} + \frac{x^{3}}{3} + \frac{12 x^{2}}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y = \frac{3 x^{4}}{3} + \frac{x^{3}}{3} + \frac{12 x^{2}}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Divide $x^{4}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{3} + \frac{12 x^{2}}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor of $12$ and $3$ .

Factor $3$ out of $12 x^{2}$ .

$y = x^{4} + \frac{x^{3}}{3} + \frac{3 (4 x^{2})}{3} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$y = x^{4} + \frac{x^{3}}{3} + \frac{3 (4 x^{2})}{3 (1)} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{3} + \frac{3 (4 x^{2})}{3 \cdot 1} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{3} + \frac{4 x^{2}}{1} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Divide $4 x^{2}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} + \frac{- 3 x^{3} \sqrt{11}}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3 x^{3} \sqrt{11}$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} + \frac{3 (- x^{3} \sqrt{11})}{3} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} + \frac{3 (- x^{3} \sqrt{11})}{3 (1)} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} + \frac{3 (- x^{3} \sqrt{11})}{3 \cdot 1} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} + \frac{- x^{3} \sqrt{11}}{1} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Divide $- x^{3} \sqrt{11}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} + \frac{- x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Move the negative in front of the fraction.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} + \frac{- 12 x \sqrt{11}}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor of $- 12$ and $3$ .

Factor $3$ out of $- 12 x \sqrt{11}$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} + \frac{3 (- 4 x \sqrt{11})}{3} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} + \frac{3 (- 4 x \sqrt{11})}{3 (1)} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} + \frac{3 (- 4 x \sqrt{11})}{3 \cdot 1} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} + \frac{- 4 x \sqrt{11}}{1} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Divide $- 4 x \sqrt{11}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} - 4 x \sqrt{11} + \frac{4 x}{3} + \frac{- 4 \sqrt{11}}{3}$

Move the negative in front of the fraction.

$y = x^{4} + \frac{x^{3}}{3} + 4 x^{2} - x^{3} \sqrt{11} - \frac{x^{2} \sqrt{11}}{3} - 4 x \sqrt{11} + \frac{4 x}{3} - \frac{4 \sqrt{11}}{3}$

Step 8

Find the Equation Given the Roots -1/3 , square root of 11 , -2i

Related Questions

Support LCMGCF.com by disabling your adblocker.