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.