Constructing a Quadratic Polynomial with Zeroes as the Squares of a Given Polynomial's Zeroes
In this article, we explore the process of constructing a quadratic polynomial whose zeroes are the squares of the zeroes of a given polynomial. We will then provide a detailed explanation and step-by-step solution for this particular case, particularly when the given polynomial is quadratic.
Introduction: The Problem Statement
Given a polynomial ( f(x) a_2x^2 a_1x a_0 ), the problem is to derive another quadratic polynomial ( h(x) b_2x^2 b_1x b_0 ) whose zeroes are the squares of the zeroes of ( f(x) ). To achieve this, we will utilize the properties of polynomial zeroes, specifically the Vieta formulas, and the symmetry of polynomials.
Vieta Formulas and Symmetric Polynomials
The Vieta formulas relate the coefficients of the polynomial to the sums and products of its zeroes. For a polynomial ( f(x) a_2x^2 a_1x a_0 ) with roots ( mu ) and ( u ), the Vieta formulas give us:
( mu u -frac{a_1}{a_2} ) ( mu u frac{a_0}{a_2} )These relationships form the foundation of our approach to constructing the polynomial ( h(x) ), whose zeroes are ( mu^2 ) and ( u^2 ).
Constructing the Quadratic Polynomial
To construct ( h(x) ) with zeroes ( mu^2 ) and ( u^2 ), we need to find the coefficients of the polynomial such that:
( mu^2 u^2 -frac{b_1}{b_2} ) ( mu^2 u^2 frac{b_0}{b_2} )Using the Vieta formulas for ( mu ) and ( u ), we can express ( mu^2 u^2 ) and ( mu^2 u^2 ) in terms of ( mu ) and ( u ):
( mu^2 u^2 (mu u)^2 - 2mu u left(-frac{a_1}{a_2}right)^2 - 2left(frac{a_0}{a_2}right) frac{a_1^2}{a_2^2} - frac{2a_0}{a_2} ) ( mu^2 u^2 (mu u)^2 left(frac{a_0}{a_2}right)^2 frac{a_0^2}{a_2^2} )Thus, the quadratic polynomial ( h(x) ) can be expressed as:
( h(x) b_2x^2 b_1x b_0 )
with coefficients determined by:
( b_2 1 ) ( b_1 left(frac{a_1^2}{a_2^2} - frac{2a_0}{a_2}right) frac{a_1^2 - 2a_0a_2}{a_2^2} ) ( b_0 frac{a_0^2}{a_2^2} )For simplicity, if we let ( b_2 a_2^2 ), then the coefficients can be simplified to:
( b_1 2a_0a_2 - a_1^2 ) ( b_0 a_0^2 )Therefore, the polynomial ( h(x) ) becomes:
( h(x) a_2^2x^2 (2a_0a_2 - a_1^2)x a_0^2 )
General Case for Any Polynomial Degree
The construction method can be generalized to polynomials of any degree. We start by expressing the new polynomial ( h(x) ) in terms of the coefficients and the Vieta formulas. The roots ( g(alpha_1), g(alpha_2), ldots, g(alpha_n) ) of ( h(x) ) are derived from the roots ( alpha_1, alpha_2, ldots, alpha_n ) of the given polynomial ( f(x) ) using the function ( g(x) ).
By the fundamental theorem of symmetric polynomials, the elementary symmetric polynomials in the roots ( g(alpha_1), g(alpha_2), ldots, g(alpha_n) ) are expressible as polynomials in the elementary symmetric polynomials of the roots ( alpha_1, alpha_2, ldots, alpha_n ). This allows us to derive the coefficients of ( h(x) ) uniquely.
Conclusion
In summary, we have demonstrated a method to construct a quadratic polynomial whose zeroes are the squares of the zeroes of a given quadratic polynomial, using the properties of Vieta formulas and the fundamental theorem of symmetric polynomials. This approach can be generalized to polynomials of any degree, providing a powerful tool in the study of polynomial equations and their roots.