To determine the number of roots of the cubic equation \( x^3 + x - 1 = 0 \), we can use the following approach:

### Fundamental Theorem of Algebra
According to the Fundamental Theorem of Algebra, a polynomial equation of degree \( n \) has exactly \( n \) roots, counting multiplicities and including complex roots. Since our equation is cubic (degree 3), it has 3 roots.

### Nature of the Roots
To determine the nature (real or complex) of the roots, we can analyze the behavior of the polynomial and its derivative:

1. **Function and Derivative Analysis**:
   \[
   f(x) = x^3 + x - 1
   \]
   \[
   f'(x) = 3x^2 + 1
   \]

   The derivative \( f'(x) = 3x^2 + 1 \) is always positive since \( 3x^2 + 1 > 0 \) for all \( x \). This implies that \( f(x) \) is a strictly increasing function.

2. **Behavior of the Function**:
   - As \( x \to -\infty \), \( f(x) \to -\infty \).
   - As \( x \to +\infty \), \( f(x) \to +\infty \).

   Since \( f(x) \) is continuous and strictly increasing, it must cross the x-axis exactly once. This guarantees one real root.

3. **Real Root and Complex Roots**:
   Given that there is one real root and a cubic equation has three roots in total, the remaining two roots must be complex conjugates. Complex roots of polynomials with real coefficients always come in conjugate pairs.

### Conclusion
The cubic equation \( x^3 + x - 1 = 0 \) has:
- 1 real root
- 2 complex conjugate roots

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:

1. How to find the complex roots of a cubic equation?
2. What is the quadratic formula and how is it derived?
3. How to solve higher-degree polynomial equations?
4. How to use the Newton-Raphson method to find roots of equations?
5. What is the Rational Root Theorem?

**Tip:** When analyzing polynomial functions, checking the first derivative can give insights into the behavior of the function, such as where it is increasing or decreasing, helping to identify the nature of its roots.

Discover how to determine the number of roots of the cubic equation x^3 + x - 1 = 0 using the Fundamental Theorem of Algebra and analysis of its derivative. This problem covers key concepts in cubic equations, polynomial functions, and complex numbers. Suitable for students in Grades 10-12.

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