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

​h(x)equals=44xSuperscript 44plus+55xcubed3plus+9494xsquared2plus+125125xnothingnegative 150−150​;

​ zero:

minus−55ii

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Part 1

The remaining zeros of h are

enter your response here.

​(Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.)

To solve this, we start by analyzing the given information:

### Function: 
\[ h(x) = 4x^4 + 5x^3 + 94x^2 + 125x - 150 \]

### Known zero: 
One of the zeros is \(-5i\).

Since the coefficients of the polynomial are real, the complex zeros must occur in conjugate pairs. Thus, if \(-5i\) is a zero, its conjugate \(5i\) is also a zero.

This gives us two zeros:
\[
x_1 = -5i, \quad x_2 = 5i
\]

### Step 1: Form a quadratic factor
The product of \((x - (-5i))(x - 5i)\) gives:
\[
(x + 5i)(x - 5i) = x^2 - (5i)^2 = x^2 + 25
\]

Thus, \(x^2 + 25\) is a factor of the polynomial.

### Step 2: Polynomial division
We now need to divide \(h(x)\) by the quadratic factor \(x^2 + 25\) to find the other factor. This will help identify the remaining zeros.

Performing synthetic or long division:

\[
h(x) \div (x^2 + 25)
\]

The result of the division is:
\[
4x^2 + 5x - 6
\]

### Step 3: Solve for remaining zeros
We now solve the quadratic \(4x^2 + 5x - 6 = 0\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 4\), \(b = 5\), and \(c = -6\).

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(4)(-6)}}{2(4)}
\]
\[
x = \frac{-5 \pm \sqrt{25 + 96}}{8}
\]
\[
x = \frac{-5 \pm \sqrt{121}}{8}
\]
\[
x = \frac{-5 \pm 11}{8}
\]

Thus, the two real roots are:
\[
x_1 = \frac{-5 + 11}{8} = \frac{6}{8} = \frac{3}{4}, \quad x_2 = \frac{-5 - 11}{8} = \frac{-16}{8} = -2
\]

### Final answer:
The remaining zeros of \(h(x)\) are:
\[
\frac{3}{4}, \, -2, \, 5i, \, -5i
\]

Would you like further clarification on any part of this solution?

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Here are five related questions:
1. What is the general form of the quadratic formula?
2. How do you find the conjugate of a complex number?
3. What is synthetic division and when is it used?
4. How can you factor a cubic polynomial?
5. What are the conditions for a polynomial to have complex roots?

**Tip**: When a polynomial has real coefficients, remember that complex roots always come in conjugate pairs!

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

h(x) = 4x^4 + 5x^3 + 94x^2 + 125x - 150

Zero: -5i

The remaining zeros of h are (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression).

Solve the polynomial equation h(x) = 4x^4 + 5x^3 + 94x^2 + 125x - 150 given one zero (-5i). Learn how to find the remaining zeros using the quadratic formula and complex conjugate root theorem. Suitable for students in Grades 10-12.

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