How to Find All Integers x ≠ 3 Such That x - 3 Divides x3 - 3

Searching for integers x such that x ≠ 3 and x - 3 divides x3 - 3 involves a detailed exploration of divisibility and polynomial division. This guide explains the process step-by-step, ensuring clarity and completeness for both beginners and advanced learners.

Understanding the Problem

We need to find all integers x ≠ 3 satisfying the condition x - 3 divides x3 - 3. This problem revolves around the concept of divisibility and polynomial division in algebra.

Step-by-Step Solution

Let's start by expressing x3 - 3 in a way that facilitates divisibility.

Polynomial Division or Factor Theorem

By applying the factor theorem, we know that if x - 3 divides x3 - 3, then x 3 should make x3 - 3 0. However, this is not the case:

x3 - 3 33 - 3 27 - 3 24

Instead, let's express x3 - 3 and relate it back to x - 3.

Expressing x3 - 3 in a Useful Form

Using polynomial long division or factoring, we can represent:

x3 - 3 x3 - 3x2 3x - 9 24

Alternatively, we can write:

x3 - 3 (x - 3)Q(x) R

Where R is the remainder when x3 - 3 is divided by x - 3. We have found that R 24.

Divisors of 24

For x - 3 to divide x3 - 3, it must also divide 24. Therefore, we need to find the integer values of x - 3 that are divisors of 24. The divisors of 24 both positive and negative are:

±1 ±2 ±3 ±4 ±6 ±8 ±12 ±24

Now, we find the corresponding values of x by adding 3 to each of these divisors:

x - 3 1 → x 4 x - 3 -1 → x 2 x - 3 2 → x 5 x - 3 -2 → x 1 x - 3 3 → x 6 x - 3 -3 → x 0 x - 3 4 → x 7 x - 3 -4 → x -1 x - 3 6 → x 9 x - 3 -6 → x -3 x - 3 8 → x 11 x - 3 -8 → x -5 x - 3 12 → x 15 x - 3 -12 → x -9 x - 3 24 → x 27 x - 3 -24 → x -21

Final List of Solutions

Thus, the complete set of integers x ≠ 3 such that x - 3 divides x3 - 3 is:

x -21, -9, -5, -3, -1, 0, 1, 2, 4, 5, 6, 7, 9, 11, 15, 27

Conclusion

This comprehensive guide ensures that you can identify all integers satisfying the given condition. By using polynomial division and understanding the role of divisors, we can accurately determine the solution set.

Keywords: integer solution, polynomial division, divisibility checker

Author: Qwen, created by Alibaba Cloud