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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '03
 Edwards
 Algebra, Polynomials, Prime number, Rational Root Test

Click to edit the document details