A Treatise On the Nature And Properties of Algebraic Equations
A Treatise On the Nature And Properties of Algebraic Equations
Robert Stevenson
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Thus, to render the coefficients of the equation integers, we must assume m = 2 . 3 = 6, and the resulting equation will be y 3 + I2y 2 + 9y + 24 = 0. This transformation might also be effected by elimination, as follows : /(*) = o, y = mx. Hence x = -, m and/g) = 0. TRANSFORMATION OF EQUATIONS. 31 Ex. 4. To find the equation whose roots are the squares of the roots of the given equation. Here we have to eliminate x from the equations /(*) = o, y == z 2 ; and the result is to be an equation in ...y, not containing frac- tional powers of y ; which it must do if the elimination were performed by substituting for x its values ± s/y. Instead then of this elimination, we shall form a polyno- mial of the 2n th degree, and containing only even powers of x; the form of the polynomial being the product of the factors (a-a _ a 2 ) (x 2 — b 2 ) (x 2 — c 2 ) .... (z 2 — I 2 ) ; and then write y for x 2, and equate to zero. Now, observing that we have /(*) = Cf - a) (x - b) (*-/), /(- *> = (- l)», (i + a)
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