To determine which of the given numbers is not a root of the polynomial [tex]\( f(x) = x^3 - 7x^2 + 14x - 8 \)[/tex], we need to evaluate the polynomial at each given candidate. A root of the polynomial is a number for which the polynomial evaluates to zero.
Given candidates: 1, 4, 2, 8
Let's evaluate [tex]\( f(x) \)[/tex] at each candidate:
1. Evaluating at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1^3 - 7 \cdot 1^2 + 14 \cdot 1 - 8
\][/tex]
[tex]\[
f(1) = 1 - 7 + 14 - 8 = 0
\][/tex]
So, [tex]\( x = 1 \)[/tex] is a root.
2. Evaluating at [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 4^3 - 7 \cdot 4^2 + 14 \cdot 4 - 8
\][/tex]
[tex]\[
f(4) = 64 - 112 + 56 - 8 = 0
\][/tex]
So, [tex]\( x = 4 \)[/tex] is a root.
3. Evaluating at [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^3 - 7 \cdot 2^2 + 14 \cdot 2 - 8
\][/tex]
[tex]\[
f(2) = 8 - 28 + 28 - 8 = 0
\][/tex]
So, [tex]\( x = 2 \)[/tex] is a root.
4. Evaluating at [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = 8^3 - 7 \cdot 8^2 + 14 \cdot 8 - 8
\][/tex]
[tex]\[
f(8) = 512 - 448 + 112 - 8 = 168
\][/tex]
Since [tex]\( f(8) = 168 \neq 0 \)[/tex], [tex]\( x = 8 \)[/tex] is not a root.
After evaluating the polynomial at all given candidates, it is clear that [tex]\( x = 8 \)[/tex] is not a root of [tex]\( f(x) = x^3 - 7x^2 + 14x - 8 \)[/tex].
Therefore, the number that is not a root of the polynomial [tex]\( f(x) = x^3 - 7x^2 + 14x - 8 \)[/tex] is [tex]\( \boxed{8} \)[/tex].
