An Introduction to the Algebra of Quantics

Cover An Introduction to the Algebra of Quantics
An Introduction to the Algebra of Quantics
E B Edwin Bailey Elliott
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Taking either root c' of this cubic we can solve the linear equations in II', Im' + l'm, mm', and so obtain their ratios, i. E. Obtain the product ll'x 2 + (lm' + l'm)xy + mm'y 2 of Ix + my, I'x + m'y but for a constant factor. /> 221. Solution of a quartic equation. When a quartic is reduced to its canonical form, or to the form a'x* + 6 cx z y z + e'y*, it is at once broken up into quadratic factors and solved. Two methods suggested by the articles which precede are here exemplified. They are
... not so simple in their use as some of those given in works on the theory of equations.
Ex. 28. By use of 220 solve the quartic equation 3o; 4 -4ar J +24ce 2 -16a; + 48 = 0.
Here a, b, c, d, e have the values 3, 1, 4, 4, 48. Thus /= 176, J = 448, and the cubic for c' is c /3 -44c / +112 = 0, of which 4 is a root. The corresponding ratios IV : lm' + I'm : mm' of 220 are 1:0: 4. The quartic has then the form a" (x + 2) 4 + J' (x - 2) 4 + 6 c" (a; 2 - 4) 2 = 0, and is in fact seen to be . E. I. E. So that the roots are + 2 1 and i (2 + 4 1 \/2).


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