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.