Solution Manual For Algebra and Trigonometry, 3rd Edition

Solutions for
Focus on Problem Solving
at www.stewartmath.com



ALGEBRA & TRIGONOMETRY 3
e

, CONTENTS
Focus on Problem Solving 1 1
Focus on Problem Solving 2 5
Focus on Problem Solving 3 9
Focus on Problem Solving 4 12
Focus on Problem Solving 5 14
Focus on Problem Solving 6 19
Focus on Problem Solving 7 22
Focus on Problem Solving 8 26
Focus on Problem Solving 9 32
Focus on Problem Solving 10 35
Focus on Problem Solving 11 38
Focus on Problem Solving 12 42
Focus on Problem Solving 13 45

,Focus on Problem Solving 1

1. Suppose that n3 − 1 is a prime number p. Then n3 − 1 = p ⇔ (n − 1)(n 2 + n + 1) = p .
Since p is a prime number one of the factors on the right-hand side of this equation
must be 1. We consider two cases.
Case 1: n − 1 = 1 . So n = 2 . But then n 2 + n + 1 = 4 + 2 + 1 = 7 , which is a prime
number.
Case 2: n 2 + n + 1 = 1 . Solving we get n = 0 . But 0 is not a natural number, so this
case is impossible.
We conclude that the only natural number n for which n3 − 1 is a prime number is the
natural number 2.



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