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To solve this, we can apply the **Remainder Theorem**. According to the Remainder Theorem, if we want to find the remainder of a polynomial \( f(x) \) when divided by \( x - a \), we evaluate \( f(a) \). In this case:

1. **Given** \( f(x) = 2x^3 - 8x^2 + 4x + 9 \).
2. We need to find the remainder when \( f(x) \) is divided by \( x - 3 \). This means we should evaluate \( f(3) \).

Let's calculate:

\[
f(3) = 2(3)^3 - 8(3)^2 + 4(3) + 9
\]

Breaking this down:

1. \( 2(3)^3 = 2 \cdot 27 = 54 \)
2. \( -8(3)^2 = -8 \cdot 9 = -72 \)
3. \( 4 \cdot 3 = 12 \)
4. The constant term is \( 9 \).

Now, combine these:

\[
f(3) = 54 - 72 + 12 + 9
\]

Calculating step-by-step:

1. \( 54 - 72 = -18 \)
2. \( -18 + 12 = -6 \)
3. \( -6 + 9 = 3 \)

So, **the remainder is 3**.

### Factor Theorem Application
The **Factor Theorem** states that \( x - a \) is a factor of \( f(x) \) if and only if \( f(a) = 0 \). Here, \( f(3) = 3 \), not 0, so **\( x - 3 \) is not a factor of \( f(x) \)**.

Would you like further details on any step?

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### Related Questions
1. How does the Remainder Theorem help simplify polynomial division?
2. What is the difference between the Remainder Theorem and the Factor Theorem?
3. How can we confirm the remainder by performing synthetic division?
4. What if we divided \( f(x) \) by a different binomial, like \( x + 2 \)?
5. Can we determine other factors of \( f(x) \) using the Factor Theorem?

**Tip:** Always double-check calculations in polynomial evaluation, especially for signs and arithmetic steps.

Use the remainder theorem to find the remainder when f(x) is divided by x - 3. Then use the factor theorem to determine whether x - 3 is a factor of f(x). f(x) = 2x^3 - 8x^2 + 4x + 9

Learn how to find the remainder when dividing the polynomial f(x) = 2x^3 - 8x^2 + 4x + 9 by x - 3 using the Remainder Theorem. Understand the application of the Factor Theorem to determine if x - 3 is a factor of f(x). Suitable for high school students studying polynomial division.

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