Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.)
P(x) = x3 + 9x2 + x + 9;     −i
x = 
 

We are given the polynomial function:

\[
P(x) = x^3 + 9x^2 + x + 9
\]

and one of its zeros, \( x = -i \).

### Step 1: Complex Conjugate
Since the coefficients of the polynomial are real, if \( -i \) is a zero, its complex conjugate \( i \) is also a zero. Therefore, \( x = i \) is another zero.

### Step 2: Factorization Using Known Zeros
We know that \( x = -i \) and \( x = i \) are zeros, so the polynomial can be divided by \( (x + i)(x - i) \), which simplifies to \( x^2 + 1 \).

Now, we can factor the polynomial by dividing \( P(x) \) by \( x^2 + 1 \).

### Step 3: Polynomial Division
We divide:

\[
\frac{x^3 + 9x^2 + x + 9}{x^2 + 1}
\]

Performing the division step-by-step:

1. \( x^3 \div x^2 = x \)
2. Multiply \( x \times (x^2 + 1) = x^3 + x \)
3. Subtract: \( (x^3 + 9x^2 + x + 9) - (x^3 + x) = 9x^2 + 9 \)
4. \( 9x^2 \div x^2 = 9 \)
5. Multiply \( 9 \times (x^2 + 1) = 9x^2 + 9 \)
6. Subtract: \( (9x^2 + 9) - (9x^2 + 9) = 0 \)

So, the quotient is \( x + 9 \), meaning:

\[
P(x) = (x^2 + 1)(x + 9)
\]

### Step 4: Find the Remaining Zero
The remaining factor is \( x + 9 \), so the remaining zero is \( x = -9 \).

### Final Answer:
The zeros of the polynomial are:
\[
x = -i, i, -9
\]

Would you like more details or explanations about any step? Here are five related questions:

1. How do you perform polynomial division in more complex cases?
2. Why must the complex conjugate \( i \) also be a zero if \( -i \) is?
3. Can you explain the significance of real and complex zeros in polynomials?
4. What happens if a polynomial has only real coefficients but a zero is complex?
5. How does the degree of a polynomial relate to the number of its zeros?

**Tip:** When working with polynomials with real coefficients, always check if complex zeros occur in conjugate pairs.

Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.) P(x) = x^3 + 9x^2 + x + 9; −i

Learn how to find the remaining zeros of the polynomial P(x) = x^3 + 9x^2 + x + 9, given one zero -i. This solution explores the use of complex conjugates, polynomial division, and factorization. Suitable for high school students (Grades 10-12).

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