Rationality Proof of Newton's Method for Finding Quadratic Trinomial Factors of Univariate Integer Coefficient Polynomials

The method of finding quadratic trinomial factors for univariate integer coefficient polynomials, proposed by the famous mathematician Isaac Newton in his mathematical monograph Arithmetica Universalis, is novel and concise, and has attracted the attention of mathematicians such as Leibniz and Bernoulli. However, no proof of this method has been given so far. This paper provides an in-depth analysis of this method and proves it with mathematical reasoning.Therefore, Newton's method of finding quadratic factors for univariate integer coefficient polynomials is reasonable, validate, and universal.