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.