Question

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

Explanation:

Step1: Recall the Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, we are dividing \( f(x) \) by \( x - 1 \), so \( a = 1 \).

Step2: Evaluate \( f(1) \)

Given \( f(x)=x^{6}-4x - 4 \), substitute \( x = 1 \) into the polynomial:
\[

$$\begin{align*} f(1)&=(1)^{6}-4(1)-4\\ &=1 - 4 - 4\\ &=1-8\\ &=- 7 \end{align*}$$

\]

Answer:

The remainder is \(-7\).