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The problem is asking whether \( x - 3 \) is a factor of the polynomial \( x^3 + 8x^2 + 9 \). To determine this, we can use the **Factor Theorem**, which states that a polynomial \( f(x) \) has \( x - c \) as a factor if and only if \( f(c) = 0 \). In this case, we are testing whether \( x - 3 \) is a factor, so we need to evaluate \( f(3) \).

### Step 1: Substitute \( x = 3 \) into the polynomial.
The polynomial is \( f(x) = x^3 + 8x^2 + 9 \).

\[
f(3) = 3^3 + 8(3^2) + 9
\]

\[
f(3) = 27 + 8(9) + 9
\]

\[
f(3) = 27 + 72 + 9 = 108
\]

### Step 2: Evaluate \( f(3) \).
Since \( f(3) = 108 \), which is **not** equal to zero, this means that \( x - 3 \) is **not** a factor of \( x^3 + 8x^2 + 9 \).

The correct answer is:
- **No, because \( f(3) = 108 \).**

Would you like further details or a synthetic division explanation?

Here are 5 related questions to extend your understanding:
1. How does the Factor Theorem help in finding factors of a polynomial?
2. What is synthetic division, and how can it be used to confirm factors?
3. Can \( x - c \) be a factor of a polynomial if \( f(c) \neq 0 \)?
4. What are the possible factors of the given polynomial \( x^3 + 8x^2 + 9 \)?
5. How can you find all real roots of a cubic polynomial?

**Tip:** The Factor Theorem is a quick way to test if a binomial is a factor, but always double-check by plugging the value into the polynomial equation.

Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first. Is x - 3 a factor of x^3 + 8x^2 + 9?

Learn how to determine if x - 3 is a factor of the polynomial x^3 + 8x^2 + 9 using the Factor Theorem. Step-by-step solution with synthetic division evaluation.

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