This problem asks us to find a number based on a condition involving its sum, its square, and its cube. We are given that the sum of these three values is 584.

Understanding the Problem Statement

Let the unknown number be represented by \(x\). According to the question, the sum of the number (\(x\)), its square (\(x^2\)), and its cube (\(x^3\)) is equal to 584. We need to find the value of \(x\) that satisfies this condition.

Setting up the Equation

Based on the problem description, we can write the condition as a mathematical equation:

\(x + x^2 + x^3 = 584\)

We need to find the value of \(x\) that makes this equation true.

Solving the Equation by Testing Options

Since we are given multiple-choice options, the easiest way to solve this equation is to test each option in the equation \(x + x^2 + x^3 = 584\) and see which one satisfies it.

Testing Option 1: If the number is 9

If \(x = 9\), the sum is:

\(9 + 9^2 + 9^3 = 9 + 81 + 729\)

\(9 + 81 + 729 = 90 + 729 = 819\)

Since \(819 \neq 584\), the number is not 9.

Testing Option 2: If the number is 8

If \(x = 8\), the sum is:

\(8 + 8^2 + 8^3 = 8 + 64 + 512\)

\(8 + 64 + 512 = 72 + 512 = 584\)

Since \(584 = 584\), this option satisfies the given condition.

Testing Option 3: If the number is 7

If \(x = 7\), the sum is:

\(7 + 7^2 + 7^3 = 7 + 49 + 343\)

\(7 + 49 + 343 = 56 + 343 = 399\)

Since \(399 \neq 584\), the number is not 7.

Testing Option 4: If the number is 6

If \(x = 6\), the sum is:

\(6 + 6^2 + 6^3 = 6 + 36 + 216\)

\(6 + 36 + 216 = 42 + 216 = 258\)

Since \(258 \neq 584\), the number is not 6.

Based on testing the options, the number that satisfies the condition \(x + x^2 + x^3 = 584\) is 8.

Summary of Calculations

The table clearly shows that only when the number is 8, the sum of the number, its square, and its cube equals 584.

Conclusion

The number that satisfies the condition is 8.

Revision Table: Number, Square, Cube Sum

Here's a quick look at how the sum changes for small integer values:

This table reinforces that 8 is the number we are looking for.

Additional Information: Solving Cubic Equations

The equation \(x + x^2 + x^3 = 584\) can be rewritten as a cubic polynomial equation:

\(x^3 + x^2 + x - 584 = 0\)

Finding the roots of a cubic equation can be complex. For general cubic equations, formulas like Cardano's method exist, but they are quite involved. When dealing with integer solutions or simple roots, methods like the Rational Root Theorem or numerical methods can be used. However, in a multiple-choice question like this, especially with integer options, testing the provided choices is usually the most straightforward and quickest approach.

In this case, since we found an integer root (x=8) that satisfies the equation, this is the desired answer. A cubic equation can have up to three roots (real or complex).