Real Root Polynomials and Real Root Preserving Transformations

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. (e fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n≥ 1 and Pn be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in Pn such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on Pn which preserve real roots of polynomials in a certain subset of Pn.


Introduction
Polynomials are the simplest and most commonly used in mathematics. We can approximate functions from real-world situation models by polynomials, and they give us results that are "close enough" to what we would get by using the actual functions and are a lot easier to use. e roots of a polynomial which are also the x-intercepts of the graph are key information when it comes to draw the polynomial graph, and they have real-world meaning when they are real numbers. Numerous approaches to compute the roots of polynomials have been developed (for example, [1][2][3]), and various efficient algorithms have been proposed (for example, [4,5]). It is natural to ask which forms of polynomials guarantee all real roots and when we map a polynomial to another polynomial, how linear transformations that preserve real roots of polynomials look like. e obvious result is proved in [6] that only nonzero multiples of the identity transformation preserve roots of all polynomials in the vector space of all polynomials of degree n or less with real coefficients, P n . e aim of this article is to provide explicit forms of polynomials in P n such that all of their roots are real as well as give explicit forms of linear transformations on P n that preserve real roots of polynomials in a certain subset of P n . e organization of this article is as follows. In Section 2, we review some basic definitions and properties about q-factorial as well as polynomials and their applications. In Section 3, we present explicit forms of polynomials and show that they always have all real roots. In Section 4, we define a subset S of P n and linear transformations on P n . en, we prove that they preserve real roots of all polynomials in S. In Section 5, we give a conclusion of our results.

Preliminary
In this section, we introduce q-factorial and recall basic theorems of polynomials and their applications.

q-Factorial
Definition 1. For nonnegative integers n and k, the number of combinations of n objects, taken k at a time, is given by By convention, n 0 is defined to be 1, and n k is defined to be 0 for n < k.
Definition 2. For a nonnegative integer n, the q-analog of n is defined to be By convention, [0] q is defined to be 0. Note that lim q⟶1 [n] q � n. (3) Example 1. Consider n � 4; we have

Polynomials
Definition 4. An element w in C is said to be a root of a given polynomial p over C if p(w) � 0. e set of all roots of p is denoted by Z(p).
Theorem 1 (the fundamental theorem of algebra; see [7]). Every nonconstant polynomial p with complex coefficients has a complex root.
is is a remarkable statement; however, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real numbers.
e following theorem which we use to prove throughout our main results provides a sufficient condition for the existence of all real roots.
ere are a variety of different applications of polynomials that we can look at. e following examples show real roots of polynomials apply to real-world situations.
Example 2. Gravity is roughly constant on the earth's surface with an acceleration of g ≈ 32 feet/second 2 , and the height of a rigid object in free fall at time t is modeled by Newton's equation of motion: where t is the elapsed time in seconds, v 0 is the initial velocity in feet/second, and p 0 is the initial height of the object above the ground level in feet. When v 0 and p 0 are known, we have . is means that the object will take t 0 seconds to hit the ground. Note that since t represents time, only nonnegative real roots apply.
In [9], the special case v 0 � 0 is used to translate height measurements of the moving object in the image to metric units in 3D world coordinates and derive relations for the case of rigid objects and then for articulated motion to estimate a person's height from the video.

Example 3.
Chemical equilibrium is a state in which the rate of the forward reaction equals the rate of the backward reaction. Consider chemical equilibrium in the gaseous system Cl 2 + 2NO � 2NOCl described by In [10], if P � 1 bar and the equilibrium constant K p � 70.23 bar at 500 K, the expression for K p can be written in the equation where x is the number of moles of A reacted at equilibrium from an initial state consisting of 1 mol of A, 2 mol of B, and 2 mol of C. Since f(x) has two complex roots and one real root x 0 ≈ 0.479764, only the real root is a chemical root of f(x).

Real Root Polynomials
Fisk [11] observed that if the coefficients of a polynomial are decreasing sufficiently rapidly, then all of the roots of the polynomial are real numbers. Motivated by this observation, we construct polynomials in the following forms and use eorem 2 to show that they have all real roots.

Proposition 1.
Let p be a polynomial in one of the following forms. en, Z(p) ⊂ R.
International Journal of Mathematics and Mathematical Sciences Similarly, if α > 2, then for 1 ≤ i ≤ n − 1, Also, if q ≥ 4, then for 1 ≤ i ≤ n − 1, [i + 1] q > q[i] q , and hence, erefore, by eorem 2, all roots of p in the above forms are real. □ Example 4. By Proposition 1, have all real roots.

Real Root Preserving Transformations
Let n ≥ 3 and S be the set of polynomials satisfying conditions in eorem 2, i.e., S � p(x) � a n x n + a n− 1 x n− 1 + · · · + a 2 x 2 + a 1 x+ a 0 ∈ P n |a i > 0 for 0 ≤ i ≤ n and a 2 i − 4a i− 1 a i+1 > 0 for 1 ≤ i ≤ n − 1}. Clearly, S is a subset of P n , and all of the roots of polynomials in S are real numbers.

Proposition 2.
e following linear transformations on P n preserve real roots of polynomials in S.
Proof. Suppose p(x) � a n x n + a n− 1 x n− 1 + · · · + a 2 x 2 + a 1 x+ a 0 is in S. en, for 1 ≤ i ≤ n − 1, we have a 2 i − 4a i− 1 a i+1 > 0. For each linear transformation, we show that it preserves real roots of p using eorem 2.