To determine whether $x-3$ is a factor of $P(x) = 2x^4 - 4x^3 - 5x^2 - 9$ using the factor theorem, we need to evaluate $P$ at $x = 3$. The factor theorem states that $x-a$ is a factor of $P(x)$ if and only if $P(a) = 0$.

Let's calculate $P(3)$:

$P(3) = 2(3)^4 - 4(3)^3 - 5(3)^2 - 9$

First, compute each term separately:

$3^4 = 81$ $2 \cdot 81 = 162$

$3^3 = 27$ $4 \cdot 27 = 108$

$3^2 = 9$ $5 \cdot 9 = 45$

Now, substitute these values back into the polynomial:

$P(3) = 162 - 108 - 45 - 9$

Perform the arithmetic step by step:

$162 - 108 = 54$ $54 - 45 = 9$ $9 - 9 = 0$

Thus, $P(3) = 0$.

Since $P(3) = 0$, by the factor theorem, $x-3$ is a factor of $P(x)$.

Would you like any further details or have any questions?

Here are 8 related questions that expand on this information:

1. What is the factor theorem and how is it used in polynomial factorization?
2. Can you show the step-by-step process for polynomial long division to verify $x-3$ as a factor of $P(x)$?
3. What are other methods to factorize a polynomial apart from using the factor theorem?
4. How do you find all factors of a given polynomial?
5. What is the difference between the factor theorem and the remainder theorem?
6. Can you use synthetic division to determine if $x-3$ is a factor of $P(x)$?
7. What happens if $P(3)$ was not zero? What would it imply about $x-3$ being a factor?
8. How do you graph a polynomial and its factors to visually confirm the roots?

Tip: When using the factor theorem, always ensure you correctly substitute the value and perform arithmetic carefully to avoid errors in determining if a polynomial has a specific factor.