The Impossibility of Having 2 Real Roots and 1 Complex Root in a Cubic Function

Cubic functions are fundamental in algebra and play a crucial role in many areas of mathematics and its applications. Understanding the nature of roots of a cubic function is critical, especially when considering the distribution of real and complex roots. This article aims to clarify why it is not possible for a cubic function to have 2 real roots and 1 complex root, following the implications of key mathematical principles.

Introduction to Cubic Functions

A cubic function is a polynomial of degree 3, following the general form:

[f(x) ax^3 bx^2 cx d]

where a, b, c, and d are real coefficients, and a ≠ 0. The degree of the polynomial determines the number of roots it can have in the complex number system.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra plays a pivotal role in understanding the roots of polynomial functions. This theorem states that any polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities. For cubic functions, this means that there are always 3 roots.

Complex Roots and Conjugate Pairs

Another important concept is that complex roots of a polynomial with real coefficients must occur in conjugate pairs. If a complex number z a bi is a root, then its conjugate (bar{z} a - bi) must also be a root. This property stems from the fact that the coefficients of the polynomial are real.

Implications for Cubic Functions

Given that complex roots must occur in conjugate pairs, let's examine what this means for a cubic function. If a cubic function has 2 real roots, the third root must be a complex root. Consequently, for complex roots to appear in pairs, this complex root must have its conjugate as another root. Therefore, it is impossible for a cubic function to have 2 real roots and 1 complex root.

The only possible scenarios for the roots of a cubic function are:

3 real roots 1 real root and 2 complex conjugate roots

This conclusion aligns with the requirement that complex roots, when present, must be in conjugate pairs due to the properties of polynomials with real coefficients.

Clarification on Complex and Real Coefficients

The previous discussion assumes that the coefficients of the cubic function are real. However, it is important to note that if the coefficients are allowed to be complex, the scenario can change. For example, consider the equation:

[x - 3i x - 3 x - 5 0]

The polynomial has complex coefficients and can have one complex root without requiring its conjugate. But for a cubic function with real coefficients, the theorem guarantees that nonreal complex roots must come in conjugate pairs.

Conclusion

Ultimately, a cubic function with real coefficients cannot have 2 real roots and 1 complex root. This is due to the requirement that complex roots appear in conjugate pairs when the coefficients are real. Therefore, the only possible combinations of roots are either all real or consisting of one real root and a pair of complex conjugates.

The key takeaways and mathematical principles commonly used in solving polynomial equations are summarized below:

Fundamental Theorem of Algebra: A polynomial of degree n has exactly n roots in the complex number system. Complex Roots Occur in Conjugate Pairs: Real coefficients imply that any complex roots must come in pairs.

By understanding these principles, one can accurately determine the root distribution of a cubic function and ensure that the solution is mathematically sound.