Find the Equation Given the Roots -4/3 , -2i

$- \frac{4}{3}$ , $- 2 i$

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

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

The factor is $x + \frac{4}{3}$

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

The factor is $x + 2 i$

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

The factor is $x - 2 i$

Combine all the factors into a single equation.

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

Multiply all the factors to simplify the equation $y = (x + \frac{4}{3}) (x + 2 i) (x - 2 i)$ .

Expand $(x + \frac{4}{3}) (x + 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Combine $\frac{4}{3}$ and $x$ .

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

Multiply $\frac{4}{3} (2 i)$ .

Combine $2$ and $\frac{4}{3}$ .

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

Multiply $2$ by $4$ .

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

Combine $\frac{8}{3}$ and $i$ .

$y = (x^{2} + 2 x i + \frac{4 x}{3} + \frac{8 i}{3}) (x - 2 i)$

Reorder the factors of $2 x i$ .

$y = (x^{2} + 2 i x + \frac{4 x}{3} + \frac{8 i}{3}) (x - 2 i)$

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

Multiply $3$ by $2$ .

$y = \frac{3 x^{2} + 6 i x + 4 x + 8 i}{3} \cdot (x - 2 i)$

Rewrite $3 x^{2} + 6 i x + 4 x + 8 i$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(3 x^{2} + 6 i x) + 4 x + 8 i}{3} \cdot (x - 2 i)$

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

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

Factor the polynomial by factoring out the greatest common factor, $x + 2 i$ .

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

Simplify terms.

Apply the distributive property.

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

Combine $\frac{(x + 2 i) (3 x + 4)}{3}$ and $x$ .

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

Multiply $\frac{(x + 2 i) (3 x + 4)}{3} (- 2 i)$ .

Combine $- 2$ and $\frac{(x + 2 i) (3 x + 4)}{3}$ .

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

Combine $\frac{- 2 ((x + 2 i) (3 x + 4))}{3}$ and $i$ .

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

Simplify terms.

Combine the numerators over the common denominator.

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

Simplify each term.

Expand $(x + 2 i) (3 x + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

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

Multiply $3$ by $2$ .

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

Multiply $4$ by $2$ .

$y = \frac{(3 x^{2} + 4 x + 6 i x + 8 i) x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{2} x + 4 x \cdot x + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Simplify.

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

Move $x$ .

$y = \frac{3 (x \cdot x^{2}) + 4 x \cdot x + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

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

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

$y = \frac{3 (x \cdot x^{2}) + 4 x \cdot x + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

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

$y = \frac{3 x^{1 + 2} + 4 x \cdot x + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Add $1$ and $2$ .

$y = \frac{3 x^{3} + 4 x \cdot x + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{3} + 4 (x \cdot x) + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x \cdot x + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i (x \cdot x) + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 ((x + 2 i) (3 x + 4)) i}{3}$

Expand $(x + 2 i) (3 x + 4)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (x (3 x + 4) + 2 i (3 x + 4)) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (x (3 x) + x \cdot 4 + 2 i (3 x + 4)) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (x (3 x) + x \cdot 4 + 2 i (3 x) + 2 i \cdot 4) i}{3}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 x \cdot x + x \cdot 4 + 2 i (3 x) + 2 i \cdot 4) i}{3}$

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

Move $x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 (x \cdot x) + x \cdot 4 + 2 i (3 x) + 2 i \cdot 4) i}{3}$

Multiply $x$ by $x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 x^{2} + x \cdot 4 + 2 i (3 x) + 2 i \cdot 4) i}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 x^{2} + 4 \cdot x + 2 i (3 x) + 2 i \cdot 4) i}{3}$

Multiply $3$ by $2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 x^{2} + 4 x + 6 i x + 2 i \cdot 4) i}{3}$

Multiply $4$ by $2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 2 (3 x^{2} + 4 x + 6 i x + 8 i) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x + (- 2 (3 x^{2}) - 2 (4 x) - 2 (6 i x) - 2 (8 i)) i}{3}$

Simplify.

Multiply $3$ by $- 2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x + (- 6 x^{2} - 2 (4 x) - 2 (6 i x) - 2 (8 i)) i}{3}$

Multiply $4$ by $- 2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x + (- 6 x^{2} - 8 x - 2 (6 i x) - 2 (8 i)) i}{3}$

Multiply $6$ by $- 2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x + (- 6 x^{2} - 8 x - 12 (i x) - 2 (8 i)) i}{3}$

Multiply $8$ by $- 2$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x + (- 6 x^{2} - 8 x - 12 (i x) - 16 i) i}{3}$

Apply the distributive property.

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 i x i - 16 i i}{3}$

Simplify.

Multiply $- 12 i x i$ .

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x (i i) - 16 i i}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x (i i) - 16 i i}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{1 + 1} - 16 i i}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{2} - 16 i i}{3}$

Multiply $- 16 i i$ .

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{2} - 16 (i i)}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{2} - 16 (i i)}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{2} - 16 i^{1 + 1}}{3}$

Add $1$ and $1$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x i^{2} - 16 i^{2}}{3}$

Simplify each term.

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i - 12 x \cdot - 1 - 16 i^{2}}{3}$

Multiply $- 1$ by $- 12$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i + 12 x - 16 i^{2}}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i + 12 x - 16 \cdot - 1}{3}$

Multiply $- 16$ by $- 1$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i + 12 x + 16}{3}$

Combine the opposite terms in $3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 x^{2} i - 8 x i + 12 x + 16$ .

Reorder the factors in the terms $6 i x^{2}$ and $- 6 x^{2} i$ .

$y = \frac{3 x^{3} + 4 x^{2} + 6 i x^{2} + 8 i x - 6 i x^{2} - 8 x i + 12 x + 16}{3}$

Subtract $6 i x^{2}$ from $6 i x^{2}$ .

$y = \frac{3 x^{3} + 4 x^{2} + 8 i x + 0 - 8 x i + 12 x + 16}{3}$

Add $3 x^{3} + 4 x^{2} + 8 i x$ and $0$ .

$y = \frac{3 x^{3} + 4 x^{2} + 8 i x - 8 x i + 12 x + 16}{3}$

Reorder the factors in the terms $8 i x$ and $- 8 x i$ .

$y = \frac{3 x^{3} + 4 x^{2} + 8 i x - 8 i x + 12 x + 16}{3}$

Subtract $8 i x$ from $8 i x$ .

$y = \frac{3 x^{3} + 4 x^{2} + 0 + 12 x + 16}{3}$

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

$y = \frac{3 x^{3} + 4 x^{2} + 12 x + 16}{3}$

Simplify the numerator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(3 x^{3} + 4 x^{2}) + 12 x + 16}{3}$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

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

Expand $(3 x + 4) (x^{2} + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$y = \frac{3 x \cdot x^{2} + 3 x \cdot 4 + 4 x^{2} + 4 \cdot 4}{3}$

Simplify each term.

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

Move $x^{2}$ .

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

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

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

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

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

$y = \frac{3 x^{2 + 1} + 3 x \cdot 4 + 4 x^{2} + 4 \cdot 4}{3}$

Add $2$ and $1$ .

$y = \frac{3 x^{3} + 3 x \cdot 4 + 4 x^{2} + 4 \cdot 4}{3}$

Multiply $4$ by $3$ .

$y = \frac{3 x^{3} + 12 x + 4 x^{2} + 4 \cdot 4}{3}$

Multiply $4$ by $4$ .

$y = \frac{3 x^{3} + 12 x + 4 x^{2} + 16}{3}$

Split the fraction $\frac{3 x^{3} + 12 x + 4 x^{2} + 16}{3}$ into two fractions.

$y = \frac{3 x^{3} + 12 x + 4 x^{2}}{3} + \frac{16}{3}$

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

$y = \frac{3 x^{3} + 12 x}{3} + \frac{4 x^{2}}{3} + \frac{16}{3}$

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

$y = \frac{3 x^{3}}{3} + \frac{12 x}{3} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y = \frac{3 x^{3}}{3} + \frac{12 x}{3} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Divide $x^{3}$ by $1$ .

$y = x^{3} + \frac{12 x}{3} + \frac{4 x^{2}}{3} + \frac{16}{3}$

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

Factor $3$ out of $12 x$ .

$y = x^{3} + \frac{3 (4 x)}{3} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$y = x^{3} + \frac{3 (4 x)}{3 (1)} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Cancel the common factor.

$y = x^{3} + \frac{3 (4 x)}{3 \cdot 1} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Rewrite the expression.

$y = x^{3} + \frac{4 x}{1} + \frac{4 x^{2}}{3} + \frac{16}{3}$

Divide $4 x$ by $1$ .

$y = x^{3} + 4 x + \frac{4 x^{2}}{3} + \frac{16}{3}$

Find the Equation Given the Roots -4/3 , -2i

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