Skip to Main Content
The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
59. Let a, b,andc be a Pythagorean triple of integers, i.e., a
2
+ b
2
= c
2
. Then the product of the three
numbers, abc, is even.
60. Fill in all the details and outline the following proof of the Rational Zero Theorem:
Let z be a rational zero of the polynomial
P ðxÞ= a
n
x
n
+ a
n1
x
n1
+
+ a
0
,
which has all integer coefficients with a
n
0 and a
0
0, with n 1. Let z = p/q be written in its lowest
terms, with q 0. Then, q divides a
n
and p divides a
0
.
Proof: By hypothesis P (z) = 0. So
a
n
p
q

n
+ a
n1
p
q

n1
+
+ a
1
p
q

+ a
0
= 0:
Therefore (why?)
a
n
p
n
+ a
n1
p
n1
q +
+ a
1
pq
n1
+ a
0
q
n
= 0: ðÞ
Thus,
a
n
p
n
= qða
n1
p
n1
+
+ a
1
pq
n2
+ a
0
q
n1
Þðwhy?Þ:
This can be rewritten as a
n
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Discrete Structures, Logic, and Computability, 4th Edition

Discrete Structures, Logic, and Computability, 4th Edition

James L. Hein

Publisher Resources

ISBN: 9780123822178