Equations, THEORY

OF. Investigation of the properties of equations of any degree in one unknown, with methods of ascertaining, exactly or approximately, the roots of numerical higher equations. The discu ion a umes f(x)=x2+A1x-1+A2x-2.....An-1X+An, in which x is a general variable, the exponents are all integral, positive, and diminish regularly by unity: the coefficient of xa is plus one and of all other terms integral. All equations having one unknown may be transformed to form f(x)=0, or made to depend upon a related function having this form. It is shown that: if f(x). be exactly divisible by (x-a), a is a root of f(x)=o: that each root of f(x)=o, when a ociated with x by a changed sign, forms a factor of f(x): f(x) being of the nth degree, f(x)=o has n. roots: no integer not a factor of the absolute term of f(x). can be a root of f(x)=o: imaginary roots occur in conjugate pairs whose sum is zero and product, positive and real: the number of po ible positive and negative roots can be determined from sequence of signs in f(x), each a ured permanence indicating a negative, each a ured variation of sign a positive, root in f(x)=o: an equation of odd degree must have one real root: of even degree and having negative absolute term, must have two real roots of opposite signs: an equation having no absolute term has one root, zero. Rules based upon these principles determine commensurable roots from the factors of the absolute term, and, through Sturm's Theorem, Horner's Method, and others, find approximately all real roots of numerical higher equations.