To form a polynomial $f(x)$ with real coefficients and the given degree and zeros, let's go step by step:

Given:

• Degree of the polynomial: 5
• Zeros: $-9$, $-i$, $8 - i$

Since the polynomial must have real coefficients, we know that non-real zeros must occur in conjugate pairs. Therefore, if $-i$ is a zero, its conjugate $i$ must also be a zero. Similarly, if $8 - i$ is a zero, its conjugate $8 + i$ must also be a zero.

Thus, the zeros of the polynomial are:

• $-9$
• $-i$ and $i$
• $8 - i$ and $8 + i$

Step 1: Write the factors corresponding to these zeros:

• For $-9$: The factor is $(x + 9)$.
• For $-i$ and $i$: The factor is $(x + i)(x - i) = x^2 + 1$.
• For $8 - i$ and $8 + i$: The factor is $(x - (8 - i))(x - (8 + i)) = (x - 8)^2 + 1 = x^2 - 16x + 65$.

Step 2: Multiply the factors:

The polynomial $f(x)$ is the product of these factors: $f(x) = (x + 9)(x^2 + 1)(x^2 - 16x + 65)$

Step 3: Expand the expression:

We first expand $(x + 9)(x^2 + 1)$: $(x + 9)(x^2 + 1) = x^3 + x + 9x^2 + 9 = x^3 + 9x^2 + x + 9$

Now multiply this by $(x^2 - 16x + 65)$: $f(x) = (x^3 + 9x^2 + x + 9)(x^2 - 16x + 65)$

You can now expand each term of $(x^3 + 9x^2 + x + 9)$ by multiplying with $(x^2 - 16x + 65)$.

Would you like me to expand it fully, or would you prefer a more compact form? Let me know how you'd like to proceed!

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Related Questions:

1. How do you identify conjugate pairs for complex zeros in polynomials?
2. What are the steps to expand a polynomial with multiple terms?
3. How can you determine the degree of a polynomial from its zeros?
4. How do you verify that a polynomial has real coefficients?
5. What is the significance of the Rational Root Theorem in finding real zeros?

Tip: When expanding polynomials, always pay attention to the structure of the terms, especially when dealing with conjugates. They often simplify the process!