Elementary Abstract Algebra- Examples and Applications, 2019a

2.1 THE ORIGIN OF COMPLEX NUMBERS 29
So far we’ve only considered square roots, but naturally we may ask the
same questions about cube roots, fourth roots, and so on:
Exercise 2.1.2. Imitate the proof of Proposition 2.1.1 to prove that −2
has no real fourth root.
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Exercise 2.1.3. Try to use the method of Proposition 2.1.1 to prove that
-4 has no real cube root. At what step does the method fail? ♦
Notice that the nth root of a is a solution of the equation x n − a =0
(and conversely–any solution of x n − a =0isannth root of a). Based on
this observation, we may generalize the notion of “root”:
Definition 2.1.4. Given a function f(x) which is defined on the real numbers
and takes real values, then a root of f(x) is any solution of the equation
f(x) =0.
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Exercise 2.1.5.
(a) Sketch the function f(x) =x 2 + 9. Does the function have any real
roots? Explain how you can use the graph to answer this question.
(b) Prove that the function f(x) =x 2 +9 has no real roots. (You may prove
by contradiction, as before).
(c) Graph the function f(x) =x 6 +7x 2 + 5 (you may use a graphing calculator).
Determine whether f(x) has any real roots. Prove your answer
(note: a picture is not a proof!).
Exercise 2.1.5 underscores an important point. A graph can be a good visual
aid, but it’s not a mathematical proof. We will often use pictures and graphs
to clarify things, but in the end we’re only certain of what we can prove.
After all, pictures can be misleading.
Exercise 2.1.6. * 3 Suppose that a · x 2n + b · x 2m + a = 0 has a real root,
where a, b, m, n are nonzero integers. What can you conclude about the
signs of a and b? Prove your answer.
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3 Asterisks (*) indicate problems that are more difficult. Take the challenge!
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