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Part 1
Use the given zero to find the remaining zeros of the function.
h left parenthesis x right parenthesis equals x Superscript 4 Baseline minus 18 x cubed plus 78 x squared plus 294 x minus 3355   ​zero: 6 minus 5 i
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Part 1
Enter the remaining zeros of h.
  
enter your response here  ​(Use a comma to separate answers as​ needed.)

We are given the polynomial function \( h(x) = x^4 - 18x^3 + 78x^2 + 294x - 3355 \) and one of its zeros, \( 6 - 5i \). Since the coefficients of the polynomial are real, the complex conjugate \( 6 + 5i \) must also be a zero.

### Steps to find the remaining zeros:

1. **Given zeros:**
   - \( 6 - 5i \)
   - \( 6 + 5i \)

2. These two zeros suggest that \( (x - (6 - 5i))(x - (6 + 5i)) = (x^2 - 12x + 61) \) is a factor of the polynomial.

3. **Divide the original polynomial** \( h(x) \) by \( (x^2 - 12x + 61) \).

Let's now divide the polynomial \( h(x) \) by \( (x^2 - 12x + 61) \) to find the remaining factor.

#### Polynomial Division

We perform the division of \( x^4 - 18x^3 + 78x^2 + 294x - 3355 \) by \( x^2 - 12x + 61 \) to obtain a quotient, which is a quadratic polynomial.

After division, we get the quotient \( x^2 - 6x - 55 \).

4. **Solve the quadratic equation** \( x^2 - 6x - 55 = 0 \) using the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-55)}}{2(1)}
\]

\[
x = \frac{6 \pm \sqrt{36 + 220}}{2} = \frac{6 \pm \sqrt{256}}{2}
\]

\[
x = \frac{6 \pm 16}{2}
\]

Thus, the two remaining zeros are:

\[
x = \frac{6 + 16}{2} = 11 \quad \text{and} \quad x = \frac{6 - 16}{2} = -5
\]

### Final Answer:
The remaining zeros of the function are:
\[
6 + 5i, \, 11, \, -5
\]

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Would you like any further details or clarifications?

Here are 5 related questions you can try:
1. What are the steps for polynomial long division?
2. How does the complex conjugate root theorem work?
3. How do you solve a quadratic equation using the quadratic formula?
4. How can you check if a polynomial division is correct?
5. Can every polynomial with real coefficients have complex roots?

**Tip:** When given a complex zero, always remember to include its conjugate when the polynomial has real coefficients.

Use the given zero to find the remaining zeros of the function. h(x) = x^4 - 18x^3 + 78x^2 + 294x - 3355  zero: 6 - 5i

Solve for the remaining zeros of the polynomial h(x) = x^4 - 18x^3 + 78x^2 + 294x - 3355 using the given zero 6 - 5i. Learn the process of polynomial division, complex conjugate root theorem, and quadratic equation solving. Ideal for students in Grades 10-12.

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