To find the remainder when [tex]\( x^3 - 2 \)[/tex] is divided by [tex]\( x - 1 \)[/tex], we use polynomial division or the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a binomial of the form [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
Here, our polynomial [tex]\( f(x) \)[/tex] is [tex]\( x^3 - 2 \)[/tex] and our binomial [tex]\( x - c \)[/tex] is [tex]\( x - 1 \)[/tex]. According to the Remainder Theorem, the remainder is found by evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex].
Let's perform this calculation:
1. Substitute [tex]\( x = 1 \)[/tex] into the polynomial [tex]\( f(x) = x^3 - 2 \)[/tex]:
[tex]\[
f(1) = 1^3 - 2
\][/tex]
2. Simplify the expression:
[tex]\[
1^3 - 2 = 1 - 2 = -1
\][/tex]
Therefore, the remainder when [tex]\( x^3 - 2 \)[/tex] is divided by [tex]\( x - 1 \)[/tex] is [tex]\(\boxed{-1}\)[/tex].
