(X.)
V. Theory of Equations.
1. In the subject “Theory of Equations” the term equation is used to denote an equation of the form xn − p1xn−1 ... ± pn = 0, where p1, p2 ... pn are regarded as known, and x as a quantity to be determined; for shortness the equation is written ƒ(x) = 0.
The equation may be numerical; that is, the coefficients p1, p2n, ... pn are then numbers—understanding by number a quantity of the form α + βi (α and β having any positive or negative real values whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting √−1.
Or the equation may be algebraical; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers.
1. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.)
Real Theory.
2. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any); that is, p1, p2, ... pn are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that an − p1an−1 ... ± pn = 0, or say, ƒ(a) = 0.
It is very useful to consider the curve y = ƒ(x),—or, what would come to the same, the curve Ay = ƒ(x),—but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinate y on a reduced scale. For instance, if the given equation be x³ − 6x² + 11x − 6.06 = 0,[1] then the curve y = x³ − 6x² + 11x − 6.06 is as shown in fig. 1, without any reduction of scale for the ordinate.
It is clear that, in general, y is a continuous one-valued function of x, finite for every finite value of x, but becoming infinite when x is infinite; i.e., assuming throughout that the coefficient of xn is +1, then when x = ∞, y = +∞; but when x = −∞, then y = +∞ or −∞, according as n is even or odd; the curve cuts any line whatever, and in particular it cuts the axis (of x) in at most n points; and the value of x, at any point of intersection with the axis, is a root of the equation ƒ(x) = 0.