Find the remainder when $\left(x^3-2\right)$ is divided by $(x-1)$.
The problem asks for the remainder when the polynomial \(P(x) = x^3 - 2\) is divided by the binomial \(x - 1\). According to the Remainder Theorem, the remainder of a polynomial division \(P(x) \div (x - c)\) is given by the value of the polynomial evaluated at \(c\), denoted as \(P(c)\). In our specific case, the divisor is \(x - 1\), which corresponds to the form \(x - c\) with \(c = 1\). Therefore, to find the remainder, we need to evaluate the polynomial at \(x = 1\).
We evaluate the polynomial \(P(x) = x^3 - 2\) at \(x = 1\):
\[P(1) = (1)^3 - 2\]
Calculating the power, we have:
\[1^3 = 1\]
Subtracting 2 from 1 gives:
\[1 - 2 = -1\]
Thus, the remainder when \(P(x) = x^3 - 2\) is divided by \(x - 1\) is \(-1\).
We have found that the remainder is \(-1\). Comparing this result with the given options:
The result \(-1\) corresponds to option (b). Therefore, the correct option is (b). The final remainder is \(\boxed{-1}\).
The Remainder Theorem is a powerful tool in algebra that allows us to find the remainder of a division without performing the long division process. For example, in computer science and data encryption, understanding how remainders work (often called modular arithmetic) helps in creating secure keys and efficiently checking data integrity. By quickly finding the remainder of a polynomial or a large expression, we can simplify complex mathematical models or verify calculations in an instant!