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To solve this problem, we need to use the **Remainder Theorem**, which states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \).

Given:
\[
f(x) = x^6 - 4x - 4
\]
We need to find the remainder when \( f(x) \) is divided by \( x - 1 \). According to the Remainder Theorem, the remainder is \( f(1) \).

Let's calculate \( f(1) \):

\[
f(1) = (1)^6 - 4(1) - 4 = 1 - 4 - 4 = -7
\]

So, the remainder when \( f(x) \) is divided by \( x - 1 \) is **-7**.

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the Remainder Theorem, and how is it used in polynomial division?
2. Can we use synthetic division to verify the remainder when dividing \( f(x) \) by \( x - 1 \)?
3. How would the remainder change if \( f(x) \) were divided by \( x - 2 \)?
4. What does the Remainder Theorem tell us about the relationship between roots and remainders?
5. How can the Remainder Theorem be applied to higher-degree polynomials?

**Tip**: When applying the Remainder Theorem, always evaluate the polynomial at \( a \) for \( x - a \), which quickly gives you the remainder without doing the full division.

If f(x) = x^6 - 4x - 4, then what is the remainder when f(x) is divided by x - 1?

Learn how to find the remainder when the polynomial f(x) = x^6 - 4x - 4 is divided by x - 1 using the Remainder Theorem. This problem covers key algebraic concepts related to polynomial division, suitable for students in Grades 9-11.

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