why is each x different from the other isnt all umbers represented by a varaible the same

The variable \( x \) in an equation represents the values that satisfy the equation. When we solve an equation, we're finding all possible values of \( x \) that make the equation true. In the given equation \( x^3 + 4x = 13x \), we found that there are multiple values of \( x \) that satisfy this equation, namely \( x = 0 \), \( x = 3 \), and \( x = -3 \).

Here's a more detailed explanation:

1. **Same Variable, Different Solutions**:
   The variable \( x \) is used to denote a number. When we solve an equation, we're finding all possible numbers (values) that \( x \) can represent to satisfy the equation. The solutions \( x = 0 \), \( x = 3 \), and \( x = -3 \) are different values that all satisfy the original equation \( x^3 + 4x = 13x \).

2. **Solving the Equation**:
   - We started with \( x^3 + 4x = 13x \).
   - We simplified it to \( x^3 - 9x = 0 \).
   - Factoring out the common term \( x \) gave us \( x(x^2 - 9) = 0 \).
   - We then factored \( x^2 - 9 \) as a difference of squares: \( x(x - 3)(x + 3) = 0 \).
   - Setting each factor equal to zero provided the solutions: \( x = 0 \), \( x = 3 \), and \( x = -3 \).

3. **Verification**:
   Each solution can be substituted back into the original equation to verify that it makes the equation true:
   - For \( x = 0 \):
     \[
     0^3 + 4 \cdot 0 = 13 \cdot 0 \implies 0 = 0 \quad \text{(True)}
     \]
   - For \( x = 3 \):
     \[
     3^3 + 4 \cdot 3 = 13 \cdot 3 \implies 27 + 12 = 39 \implies 39 = 39 \quad \text{(True)}
     \]
   - For \( x = -3 \):
     \[
     (-3)^3 + 4 \cdot (-3) = 13 \cdot (-3) \implies -27 - 12 = -39 \implies -39 = -39 \quad \text{(True)}
     \]

So, while the variable \( x \) is the same, it can represent different values that satisfy the equation. In this case, \( x \) can be 0, 3, or -3.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do polynomial equations differ from linear equations in terms of solutions?
2. Can a polynomial equation have complex (non-real) solutions?
3. What is the geometric interpretation of the solutions of a polynomial equation?
4. How can graphing help in understanding the solutions of a polynomial equation?
5. Are there polynomial equations that have only one unique solution?
6. How does the degree of a polynomial relate to the number of solutions?
7. What is the significance of multiple roots in polynomial equations?
8. How do we handle equations with higher-degree polynomials (quartic, quintic, etc.)?

**Tip:** Always verify your solutions by substituting them back into the original equation to ensure they satisfy it.

Explore why each 'x' in equations can represent different values. Learn with detailed examples and explanations on how variables like 'x' can have multiple solutions in polynomial equations. Suitable for Grades 9-12.

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