New Perspectives on the Roots of Real Polynomials of Degree n and Number Theory

In this work we obtained some results on the real and complex roots of real polynomials of the form \({P{(x)}} = {x^{n} + {a_{n - 1}x^{n - 1}} + \cdots + {a_{1}x} + a_{0}}\), where the \(a_{k}\) are real numbers for \(k \in {\{ 0,\ldots,{n - 1}\}}\), we also obtained results on linear Diophantine equations of the form \({{ax} + {by} + {cz}} = d\), where \(a,b,c\) and \(d\) are integers. To obtain the desired results, the relation that states the following was used: for any real numbers \(A\) and \(B\), there exists a unique real number \(\lambda\), such that \({A + B} = {\lambda{\lbrack{A^{2} + B^{2}}\rbrack}}\). This result is appropriately linked to the object of study. For the polynomials, optimal domains were obtained where the real or complex roots are found, without the use of higher calculus. For the linear Diophantine equation, the desired solutions were obtained, which were found by establishing several links between the Diophantine equation and the relationship \({A + B} = {\lambda{\lbrack{A^{2} + B^{2}}\rbrack}}\). Several examples of the results obtained are illustrated, which are intended to show the benefits of this proposal. Additionally, we obtained the solution of Fermat’s last Theorem in an elegant, simple and unprecedented way, different from what has been done by other authors.