Use the given zero to find the remaining zeros of the function.

​h(x)equals=33xSuperscript 44plus+77xcubed3plus+66xsquared2plus+2828xnothingnegative 24−24​;

​ zero:

minus−22ii

We are given the function:

\[
h(x) = 3x^4 + 7x^3 + 6x^2 + 28x - 24
\]

with one known zero \( -2i \).

### Step 1: Use Complex Conjugate Root Theorem
The coefficients of the polynomial are real, meaning that if \( -2i \) is a zero, its conjugate \( 2i \) must also be a zero.

So far, we have two zeros: 
\[
-2i \quad \text{and} \quad 2i.
\]

### Step 2: Factor out the quadratic corresponding to the complex roots

The factor that corresponds to the complex roots \( -2i \) and \( 2i \) is:
\[
(x - (-2i))(x - 2i) = (x + 2i)(x - 2i) = x^2 + 4.
\]

Thus, \( h(x) \) can be factored as:
\[
h(x) = (x^2 + 4)q(x),
\]
where \( q(x) \) is the remaining quadratic factor that we need to determine.

### Step 3: Polynomial Division

We will now divide \( h(x) \) by \( x^2 + 4 \) to find \( q(x) \).

Using synthetic or long division of the polynomial \( 3x^4 + 7x^3 + 6x^2 + 28x - 24 \) by \( x^2 + 4 \), we get:

\[
q(x) = 3x^2 + 7x - 6.
\]

### Step 4: Solve for the remaining zeros

Now, we solve the quadratic \( 3x^2 + 7x - 6 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \( a = 3 \), \( b = 7 \), and \( c = -6 \).

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(3)(-6)}}{2(3)} = \frac{-7 \pm \sqrt{49 + 72}}{6} = \frac{-7 \pm \sqrt{121}}{6} = \frac{-7 \pm 11}{6}.
\]

Thus, the two solutions are:
\[
x = \frac{-7 + 11}{6} = \frac{4}{6} = \frac{2}{3}
\]
and
\[
x = \frac{-7 - 11}{6} = \frac{-18}{6} = -3.
\]

### Step 5: Final zeros

The zeros of the polynomial \( h(x) \) are:
\[
-2i, \, 2i, \, \frac{2}{3}, \, -3.
\]

Would you like more details or have any questions?

Here are five related questions for further practice:
1. What are the roots of \( 4x^4 - 64x^2 + 256 = 0 \)?
2. How does the complex conjugate root theorem apply to real-coefficient polynomials?
3. How do you factor a polynomial given complex roots?
4. What is the general approach to solving higher-degree polynomials?
5. How can synthetic division be applied to find polynomial quotients efficiently?

**Tip:** When solving polynomial equations, always check for complex conjugate pairs if the coefficients are real, as this can simplify the process.

Use the given zero to find the remaining zeros of the function h(x) = 3x^4 + 7x^3 + 6x^2 + 28x - 24; zero: -2i

Explore how to find the remaining zeros of the polynomial h(x) = 3x^4 + 7x^3 + 6x^2 + 28x - 24, given the complex root -2i. Learn about the application of the Complex Conjugate Root Theorem, polynomial division, and solving quadratic equations. Suitable for high school students.

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