Math 302
Modern Algebra
Spring 2009
Homework Solutions
4.5
IRREDUCIBILITY IN
Q
[
x
]
1.
Use the Rational Root Test to write each polynomial as a product of irreducible polynomials in
Q
[
x
].
(e)
2
x
4
+ 7
x
3
+ 5
x
2
+ 7
x
+ 3
By the Rational Root Test, if
r/s
is a rational root, then
r

3 and
s

2, so
r
=
±
1
,
±
3
and
s
=
±
1
,
±
2
.
Therefore the possible rational roots are
r/s
=
±
1
2
,
±
1
,
±
3
2
,
±
3
.
Testing the roots one by one, we Fnd that
−
1
2
and
−
3 are roots and the polynomial can be
factored as
2
x
4
+ 7
x
3
+ 5
x
2
+ 7
x
+ 3 = 2(
x
+
1
2
)(
x
+ 3)(
x
2
+ 1)
.
Since
x
+
1
2
and
x
+ 3 are linear, then both are irreducible in
Q
[
x
]. Since
x
2
+ 1 has no rational
roots, then it’s irreducible in
Q
[
x
] by Corollary 4.18.
(f)
6
x
4
−
31
x
3
+ 25
x
2
+ 33
x
+ 7
By the Rational Root Test, if
r/s
is a rational root, then
r

7 and
s

6, so
r
=
±
1
,
±
7
and
s
=
±
1
,
±
2
,
±
3
,
±
6
.
Therefore the possible rational roots are
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 Spring '03
 Edwards
 Algebra, Polynomials

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