Sure, let's discuss how Bart can use the factor theorem to determine if [tex]\( x-4 \)[/tex] is a factor of the polynomial [tex]\( 3x^4 - 10x^3 + 11x - 172 \)[/tex].
The factor theorem states that [tex]\( x-c \)[/tex] is a factor of a polynomial [tex]\( f(x) \)[/tex] if and only if [tex]\( f(c) = 0 \)[/tex]. In this specific problem, Bart needs to check if [tex]\( x-4 \)[/tex] is a factor, so he will substitute [tex]\( x = 4 \)[/tex] into the polynomial [tex]\( 3x^4 - 10x^3 + 11x - 172 \)[/tex].
Now, we follow the steps below:
1. Recall the polynomial [tex]\( f(x) = 3x^4 - 10x^3 + 11x - 172 \)[/tex].
2. Substitute [tex]\( x = 4 \)[/tex] into the polynomial:
[tex]\[
f(4) = 3(4)^4 - 10(4)^3 + 11(4) - 172
\][/tex]
3. Calculate [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = 3 \cdot 256 - 10 \cdot 64 + 44 - 172
\][/tex]
[tex]\[
f(4) = 768 - 640 + 44 - 172
\][/tex]
[tex]\[
f(4) = 768 - 640 + 44 - 172 = 0
\][/tex]
4. Since [tex]\( f(4) = 0 \)[/tex], by the factor theorem, [tex]\( x-4 \)[/tex] is indeed a factor of [tex]\( 3x^4 - 10x^3 + 11x - 172 \)[/tex].
Therefore, Bart evaluates [tex]\( 3x^4 - 10x^3 + 11x - 172 \)[/tex] at [tex]\( x = 4 \)[/tex] and determines the value of the expression is [tex]\( 0 \)[/tex]. Consequently, Bart correctly concludes that [tex]\( x-4 \)[/tex] is a factor.
So, the correct statement is:
Bart evaluates [tex]\( 3x^4 - 10x^3 + 11x - 172 \)[/tex] when [tex]\( x = 4 \)[/tex]. He determines that the value of the expression is [tex]\( 0 \)[/tex], and concludes that [tex]\( x-4 \)[/tex] is a factor.
