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The given problem asks to find the remainder when the polynomial \( f(x) = 3x^5 + 4x^3 - 4 \) is divided by \( x - 2 \).

To solve this, we can use the **Remainder Theorem**, which states that the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is simply \( f(a) \).

Here, \( f(x) \) is divided by \( x - 2 \), so the remainder is \( f(2) \). Let’s calculate \( f(2) \):

\[
f(2) = 3(2)^5 + 4(2)^3 - 4
\]

Now, I will compute that for you.The remainder when \( f(x) = 3x^5 + 4x^3 - 4 \) is divided by \( x - 2 \) is \( 124 \).

Would you like more details or have any further questions? 

Here are 5 related questions to explore:
1. How does the Remainder Theorem apply to other polynomials?
2. What is the remainder if the same polynomial is divided by \( x + 3 \)?
3. How do you use synthetic division to verify the remainder?
4. What happens to the remainder if we divide by \( x + a \) instead of \( x - a \)?
5. How do we factor polynomials using the Factor Theorem?

**Tip:** The Remainder Theorem is closely related to the Factor Theorem, which tells you whether a polynomial is divisible by \( x - a \). If the remainder is 0, then \( x - a \) is a factor!

If f(x) = 3x^5 + 4x^3 - 4, then what is the remainder when f(x) is divided by x - 2?

Learn how to find the remainder when the polynomial f(x) = 3x^5 + 4x^3 - 4 is divided by x - 2 using the Remainder Theorem. This algebraic problem involves polynomials and their division, making it suitable for high school students (Grades 10-12).

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