Use the factor theorem to determine whether x-3 is a factor of $$ P\left(x\right)=2x^4-4x^3-5x^2-9 $$ Specifically, evaluate P at the proper value, and then determine whether x-3 is a factor.

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.

Learn how to use the factor theorem to determine whether x-3 is a factor of the polynomial P(x) = 2x^4 - 4x^3 - 5x^2 - 9. Step-by-step solution with polynomial evaluation and theorem application. Suitable for high school students studying polynomial factorization and the factor theorem.

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