Problem 36 - A Textbook of Differential Calculus

Question

If \(a_{1} x^{3}+b_{1} x^{2}+c_{1} x+d_{1}=0\) and \(a_{2} x^{3}+b_{2} x^{2}+c_{2} x+d_{2}=0\) have a pair of repeated common roots, then $$ \left|\begin{array}{ccc} 3 a_{1} & 2 b_{1} & c_{1} \\ 3 a_{2} & 2 b_{2} & c_{2} \\ a_{2} b_{1}-a_{1} b_{2} & c_{1} a_{2}-c_{2} a_{1} & d_{1} a_{2}-d_{2} a_{1} \end{array}\right| \text { is } $$ (a) 0 (b) \(c_{1} a_{2}-c_{2} a_{1}\) (c) \(a_{1} b_{2}-a_{2} b_{1}\) (d) \(d_{1} a_{2}-d_{2} a_{1}\)

Step-by-Step Solution

Answer

The determinant is 0, matching option (a).

Step 1: Identify the Problem Setup

The problem involves two cubic polynomials that share a pair of repeated common roots. This means both polynomials have at least one repeated root, say \( \alpha \), that is common to both equations. The task is to determine under what condition this configuration leads to the determinant of a given matrix being zero.

Step 2: Understand Shared Roots and Determinants

Since the polynomials share a pair of repeated roots \( \alpha \), this effectively means \( (x-\alpha)^2 \) is a factor of both polynomials. For such setups with repeated roots, the determinant of a matrix constructed from the coefficients must be zero to satisfy this condition (general result from resultant theory).

Step 3: Set Up the Matrix and Calculate its Determinant

The matrix given is \[ \begin{vmatrix} 3 a_{1} & 2 b_{1} & c_{1} \ 3 a_{2} & 2 b_{2} & c_{2} \ a_{2} b_{1}-a_{1} b_{2} & c_{1} a_{2}-c_{2} a_{1} & d_{1} a_{2}-d_{2} a_{1} \end{vmatrix} \]. Calculate its determinant using expansion by the first row: \[ D = 3a_1 \begin{vmatrix} 2b_2 & c_2 \ c_1a_2 - c_2a_1 & d_1a_2 - d_2a_1 \end{vmatrix} - 2b_1 \begin{vmatrix} 3a_2 & c_2 \ a_2b_1 - a_1b_2 & d_1a_2 - d_2a_1 \end{vmatrix} + c_1 \begin{vmatrix} 3a_2 & 2b_2 \ a_2b_1 - a_1b_2 & c_1a_2 - c_2a_1 \end{vmatrix} \]

Step 4: Simplify the Determinant Expressions

Perform the calculations for each of the smaller 2x2 determinants and simplify:- For \( 3a_1 \begin{vmatrix} 2b_2 & c_2 \ c_1a_2 - c_2a_1 & d_1a_2 - d_2a_1 \end{vmatrix}\), the result is \( 3a_{1}(2b_2(d_1a_2 - d_2a_1) - c_2(c_1a_2 - c_2a_1)) \).- For \(-2b_1\begin{vmatrix} 3a_2 & c_2 \ a_2b_1 - a_1b_2 & d_1a_2 - d_2a_1 \end{vmatrix}\), the result is \(-2b_{1}(3a_2(d_1a_2 - d_2a_1) - c_2(a_2b_1 - a_1b_2)) \).- For \(c_1\begin{vmatrix} 3a_2 & 2b_2 \ a_2b_1 - a_1b_2 & c_1a_2 - c_2a_1 \end{vmatrix}\), the result is \(c_{1}(3a_2(c_1a_2 - c_2a_1) - 2b_2(a_2b_1 - a_1b_2)) \).Combine all these to check for cancellation.

Step 5: Conclusion Based on Determinant Being Zero

After simplifying each part of the expression and adding them, notice that for the determinant to be zero, certain terms need to cancel out completely. This typically happens when specific conditions on the coefficients hold, like a resultant or generalized form of symmetric property implied by the problem statement. Given the choices (a) through (d), if the solution to these conditions leads to simplicity, the choice (a) as zero is plausible.

Key Concepts

Repeated Roots

Repeated roots in polynomial equations occur when a specific root appears more than once. This is significant in algebra because repeated roots indicate a higher multiplicity, such as a double or triple root. When two cubic equations share a repeated root, it means there is a common solution that satisfies the derivative as well. For instance, if the root is \(\alpha\), it appears as \((x-\alpha)^2\) in the factorization.
We often encounter repeated roots in the context of complex polynomial equations where higher multiplicity plays a crucial role in how solutions behave. Understanding repeated roots is crucial as it affects the stability and the configuration of polynomial graphs.
When dealing with matrices that represent these equations, finding repeated roots is essential in determining the conditions for a zero determinant, especially when leveraging resultant theory.

Determinants

The determinant of a matrix is a special number that can be calculated from its elements and is pivotal in linear algebra. It is used to determine whether a system of linear equations has a unique solution. If the determinant is zero, the system does not have a unique solution, which aligns with having repeated roots.
In this context, the determinant helps in establishing the condition under which two cubic equations have repeated common roots. This condition results in the determinant of a specially constructed matrix being zero. Calculating determinants often involves the use of cofactors and minors, expanding by rows or columns, which theory was notably used in our step-by-step solution.
The determinant's value can inform about the properties of the linear transformation associated with the matrix, such as scaling effects or whether the transformation preserves orientation.

Polynomial Roots

The roots of a polynomial are the solutions to the equation when set equal to zero, essentially the values of \(x\) that satisfy the equation. These roots are important as they represent the x-values where the polynomial crosses or touches the x-axis.
In the case of cubic polynomials like ours, finding the roots is essential for understanding intersections with another polynomial, especially when those roots are repeated. The roots serve as critical points in the behavior of the polynomial function itself.
By using algebraic techniques and sometimes calculus for higher-degree polynomials, roots can be derived, allowing for insights into the structure and nature of solutions of the equation. The behavior of polynomial roots is very well captured through examining their multiplicities and the positions they occupy in the factorized polynomial.

Matrix Determinant

A matrix determinant is a computed value from a square matrix, which determines whether the matrix is invertible, as well as linking closely to the area (in two dimensions) or volume (in higher dimensions) transformations by that matrix. Specifically, for a 3x3 matrix as in our exercise, it is calculated using the rule of Sarrus or cofactor expansion.
The significance of the matrix determinant in this kind of exercise arises from its relation to the conditions ensuring repeated roots between polynomials, where setting the determinant to zero implies dependent equations with shared repeated roots. This showcases how matrix algebra enables problem-solving in polynomial contexts.
Understanding the computation and interpretation of the matrix determinant helps not just in theoretical exercises but also in practical applications, including computer graphics, physics simulations, and solving simultaneous equations.

Resultant Theory

Resultant theory provides a powerful algebraic tool used to eliminate variables between polynomial equations, thus offering a condition for the existence of common roots. In simpler terms, it's a technique to determine when two polynomials share common roots without finding them explicitly.
Our exercise utilizes this theory by focusing on the determinant that emerges from combining matrices of polynomial coefficients. The matrix is thus central to the theory, and its zero determinant condition indicates shared roots.
Resultant theory is often used in advanced algebra to solve real-world problems such as signal processing or in systems requiring solutions for multiple polynomials simultaneously. Knowing how resultant theory applies offers a deeper insight into algebraic structures and solutions often used in engineering and technology.