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We prove new theorems for the polynomial expansions of $x^n \pm y^n$ in termsof the binary quadratic forms $\alpha x^2 + \beta xy + \alpha y^2 $ and $a x^2+ bxy + a y^2 $. The paper gives new arithmetic differential approach tocompute the coefficients. Also, the paper gives generalization to well-knownpolynomial identity in the history of number theory. The paper highlights theemergence of a new class of polynomials that unify many well-known sequencesincluding the Chebyshev polynomials of the first and second kind, Dicksonpolynomials of the first and second kind, Lucas and Fibonacci numbers, Mersennenumbers, Pell polynomials, Pell-Lucas polynomials, and Fermat numbers. Also,this paper highlights the emergence of the notions of trajectories and orbitsof certain integers that passes through many well-known polynomials andsequences. The Lucas-Fibonacci trajectory, the Lucas-Pell trajectory, theFibonacci-Pell trajectory, the Fibonacci-Lucas trajectory, theChebyshev-Dickson trajectory of the first kind, the Chebyshev-Dicksontrajectory of the second kind, and others are new trajectories included in thispaper. Also, the Lucas orbit, Fibonacci orbit, Mersenne orbit, Lucas-Fibonacciorbit, Fermat orbit, and others are new orbits included in this paper.

New expansions for $x^n \pm y^n$ in terms of quadratic forms