Question

use the factor theorem to determine if (x - 4) is a factor of ( x^5 - 3x^4 - x - 3 ). (1 point) \\( \bigcirc \\) (x - 4) is a factor because ( f(4) = 0 ). \\( \bigcirc \\) (x - 4) is a factor because ( f(4) \
eq 0 ). \\( \bigcirc \\) (x - 4) is not a factor because ( f(4) = 0 ). \\( \bigcirc \\) (x - 4) is not a factor because ( f(4) \
eq 0 ).

Explanation:

Step1: Recall Factor Theorem

The Factor Theorem states that a polynomial \( f(x) \) has a factor \( (x - c) \) if and only if \( f(c)=0 \). Here, \( c = 4 \) (since the factor is \( (x - 4) \)), so we need to evaluate \( f(4) \) for the polynomial \( f(x)=x^{5}-3x^{4}-x - 3 \).

Step2: Substitute \( x = 4 \) into \( f(x) \)

\[

$$\begin{align*} f(4)&=4^{5}-3\times4^{4}-4 - 3\\ &=1024-3\times256-4 - 3\\ &=1024 - 768-4 - 3\\ &=256-4 - 3\\ &=252 - 3\\ &=249 \end{align*}$$

\]
Since \( f(4)=249
eq0 \), by the Factor Theorem, \( (x - 4) \) is not a factor of \( f(x) \) because \( f(4)
eq0 \).

Answer:

D. \((x - 4)\) is not a factor because \( f(4)
eq0 \)