Question

Use the Factor Theorem to determine if left parenthesis x minus 4 right parenthesis is a factor of x superscript 5 baseline minus 3 x superscript 4 baseline minus x minus 3.
(1 point)
Responses

left parenthesis x minus 4 right parenthesis is a factor because f left parenthesis 4 right parenthesis equals 0.
Image with alt text: left parenthesis x minus 4 right parenthesis is a factor because Image with alt text: f left parenthesis 4 right parenthesis equals 0 .

left parenthesis x minus 4 right parenthesis is a factor because f left parenthesis 4 right parenthesis not equals 0.
Image with alt text: left parenthesis x minus 4 right parenthesis is a factor because Image with alt text: f left parenthesis 4 right parenthesis not equals 0 .

left parenthesis x minus 4 right parenthesis is not a factor because f left parenthesis 4 right parenthesis equals 0.
Image with alt text: left parenthesis x minus 4 right parenthesis is not a factor because Image with alt text: f left parenthesis 4 right parenthesis equals 0 .

left parenthesis x minus 4 right parenthesis is not a factor because f left parenthesis 4 right parenthesis not equals 0.

Answer 1

To use the Factor Theorem, we first need to evaluate the polynomial \( f(x) = x^5 - 3x^4 - x - 3 \) at \( x = 4 \):

1. Calculate \( f(4) \):

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

Calculating each term:

- \( 4^5 = 1024 \)
- \( 4^4 = 256 \), thus \( 3(4^4) = 3 \times 256 = 768 \)

Now substituting back into the function:

\[
f(4) = 1024 - 768 - 4 - 3 = 1024 - 768 - 7 = 249
\]

Since \( f(4) \neq 0 \), according to the Factor Theorem, \( (x - 4) \) is **not a factor** of the polynomial.

Thus, the correct response is:

**left parenthesis x minus 4 right parenthesis is not a factor because f left parenthesis 4 right parenthesis not equals 0.**