If x is a whole number (including zero), then the expression x²(x² − 1) is always divisible by which of the following numbers?

Introduction:
This is a number theory question about divisibility properties of an algebraic expression that depends on a whole number x. You must determine a constant divisor that will always divide x²(x² − 1) for any whole number x. This requires factoring and reasoning about consecutive integers and their divisibility by 2, 3 and 4.

Given Data / Assumptions:

• x is a whole number (0, 1, 2, 3, ...).
• The expression is x²(x² − 1).
• We must find which number from the options 4, 8, 12 or None is a guaranteed divisor for all such x.

Concept / Approach:
First, factor the expression:
x²(x² − 1) = x²(x − 1)(x + 1).This is a product involving x, x − 1 and x + 1, which are three consecutive integers, together with an extra x. We then analyze factors of 2, 3 and 4 in this product and decide the largest fixed divisor among the given options.

Step-by-Step Solution:
Step 1: Factor the expression.x²(x² − 1) = x²(x − 1)(x + 1).Step 2: Note the structure.(x − 1), x and (x + 1) are three consecutive integers.Step 3: Divisibility by 3.Among any three consecutive integers, one is divisible by 3, so the product (x − 1)x(x + 1) is always divisible by 3. Therefore x²(x² − 1) is divisible by 3.Step 4: Divisibility by 4.If x is even, then x² has at least a factor 4.If x is odd, then (x − 1) and (x + 1) are consecutive even numbers, one of which is divisible by 4. Thus (x − 1)(x + 1) has a factor 4.In either case, the whole expression has a factor 4.Step 5: Combine results.We have at least one factor 3 and one factor 4 in the product, so the expression is always divisible by 3 * 4 = 12.

Verification / Alternative check:
Test with small whole numbers: x = 2 ⇒ x²(x² − 1) = 4 * 3 = 12, divisible by 12. x = 3 ⇒ 9 * 8 = 72, divisible by 12. x = 4 ⇒ 16 * 15 = 240, divisible by 12. In each case, 12 divides the expression, and the reasoning above shows this holds for all whole numbers.

Why Other Options Are Wrong:
4 is a divisor but not the greatest among the options; 12 is stronger and also always divides. 8 fails for x = 2 because 12 is not divisible by 8. “None” is incorrect since we have found a valid fixed divisor, and 6 is too small given that 12 always works.

Common Pitfalls:
Some students do not factor the expression and instead test a few values of x, which may not be convincing. Others overlook the behavior of consecutive integers regarding divisibility by 2, 3 and 4. Recognizing patterns in consecutive numbers is essential here.

Final Answer:
The expression x²(x² − 1) is always divisible by 12 for any whole number x.