Determining the Nature of Roots of a Cubic Equation Without Explicit Solutions

When faced with a cubic polynomial, often the primary objective is to ascertain the nature and number of its real roots, without actually solving the equation. This capability is particularly valuable in theoretical contexts or when the coefficients of the polynomial are complex or inconvenient to handle directly. In this article, we explore a method to determine the nature of the roots of a cubic equation without solving it explicitly.

Introduction to the Problem

Consider a cubic polynomial with a non-zero leading coefficient. By dividing by this coefficient, we can work with a cubic polynomial of the form:

[ a(x - x_0)^3 b(x - x_0)^2 c(x - x_0) d 0 ]

where (a, b, c,) and (d) are real coefficients and (x_0) is a constant. This transformation does not affect the number or nature of the roots; it merely shifts their location on the real line.

Depressed Cubic Polynomial

We define a “depressed” cubic polynomial as one for which there is no (x^2) term:

[ x^3 px q 0 ]

where (p) and (q) are the coefficients of (x) and the constant term, respectively. The nature and number of the real roots of this depressed cubic polynomial can be determined by examining the sign of a specific expression involving (p) and (q).

Theorem and Proof

The following theorem provides a straightforward method to determine the nature of the roots of the depressed cubic equation:

Theorem: The number of real roots of the distinct depressed cubic equation (x^3 px q 0), with both (p eq 0) and (q eq 0), is determined by the sign of the discriminant (Delta p^3 27q^2) as follows:

There are exactly three distinct real roots if and only if (Delta > 0). In this case, each one of the following three disjoint open intervals contains exactly one root: There are exactly two distinct real roots if and only if (Delta There is a single real root if and only if (Delta 0). In this scenario, the single root lies outside the closed interval ([-|q|/2p, |q|/2p]).

Proof:

Part i: When there are three distinct real roots

Suppose the given depressed cubic polynomial has three distinct real roots. Then we can factorize it as follows:

[ (x - alpha)(x - beta)(x - gamma) 0 ]

where (alpha, beta,) and (gamma) are the three distinct real roots. By expanding this product, we get:

[ x^3 - (alpha beta gamma)x^2 (alphabeta betagamma gammaalpha)x - alphabetagamma 0 ]

Comparing this with (x^3 px q 0), we find that (alpha beta gamma 0) (since the coefficient of (x^2) is 0) and (p alphabeta betagamma gammaalpha) and (-q alphabetagamma). By Viète's formulas, the discriminant (Delta) of the cubic polynomial can be shown to be:

[ Delta p^3 27q^2 (alphabeta betagamma gammaalpha)^3 27(-alphabetagamma)^2 (alphabeta betagamma gammaalpha)^3 27(alphabetagamma)^2 ]

Given that (p eq 0), (Delta > 0).

To show that (Delta > 0) is a necessary condition, consider the factorization of the cubic polynomial when (Delta > 0). By the intermediate value theorem, the polynomial must have three distinct real roots, as the intermediate values of the polynomial at the interval boundaries guarantee at least one root in each subinterval.

Part ii: When there are two distinct real roots

Suppose the given depressed cubic polynomial has exactly two distinct real roots and one repeated root. Without loss of generality, let the roots be (alpha, alpha, beta). Then we can write:

[ (x - alpha)(x - alpha)(x - beta) 0 ]

Expanding this, we get:

[ x^3 - 2alpha x^2 (alpha^2 beta)x - alpha^2beta 0 ]

Comparing with (x^3 px q 0), we find that (2alpha 0), which is not possible (since (alpha eq 0)). Therefore, consider the quadratic factor with a double root:

[ p alpha^2 beta quad text{and} quad -q alpha^2beta ]

Given that (p eq 0), we have (Delta

Conversely, if (Delta

Part iii: When there is a single real root

If there is a single real root, then the cubic polynomial has a double root and a simple root. The discriminant (Delta 0) implies that the roots are such that the interval ([-|q|/2p, |q|/2p]) contains no roots. This is because the interval boundaries are the points where the polynomial changes sign, and the single real root lies outside this interval.

Conclusion

By analyzing the discriminant (Delta p^3 27q^2) of a depressed cubic polynomial, we can efficiently determine the nature and number of its real roots. This theorem provides a powerful tool for understanding the behavior of cubic polynomials without the need to explicitly solve the equation. The results have profound implications in algebra, number theory, and applied mathematics, offering a concise method to explore the root structure of cubic equations.

Happy exploring the world of cubic polynomials!