An introduction to the modern theory of equations, by Florian Cajori.

THEORY OF EQUATIONS for quadratic rational factors, divide x" + a(x" - + c2 ~- a3x + a4 by x2 + ax -+ /, where a and P are integers to be determinied, if possible. That there may be no remainder, we must have a3 - aL/ + a3 = a(a2 - /3 - a + a2), a4 = P(a2- / - aa + C2). 1 Hence a=c = - II c4 - /2 We have the rule: See whether any factor f1 of a4 makes a an integer in II. If a cand 3 are su7ch integers, which also satisfy I, then x2 +- ax + t is a rational factor sought. Similarly, if f(x) is of the fifth degree. First search for linear rational factors x - c. If none are present, there is no quartic rational factor. Look for a quadratic rational factor x2 + ax + /3. If quadratic factors are likewise absent, there can be no cubic rational factor, and the function is irreducible. Dividing xi + C4tx4 +- ax3 +- a3x2 +- c4x - a5 by x2 +- ax +- /, we get as the conditions for zero remainder, a4 - Cf2 + /2 + (1/ -_ 2 = a(a - aC/l3 + 2 a/3 - a2a + a(1(2 - a3), a5 = 3(as - a 2/3 + 2 a/ - a - + a ca2 - ). III Whence a =,-c1-4cc 2 co where co = /2, C1l= C5 - al2, c2 = 2 a - a4 - /3. If / is a factor of as, if a is an integer, and III is satisfied, then x2 + ax 4- is a factor sought. Ex. 1. Is f(x) -5 x+4 4 + 4 x3 + 9 x2 + 8 x + 2 reducible in 9()? Since f(x) does not vanish for x = ~ 1 or + 2, there are no linear nor quartic factors in 92(). Take, =- 2, then co = 4, cl - - 14, c2= -8, a=4. Condition III is satisfied. Hence x2 + 4: + 2 is a factor.

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 130
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2025.
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