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Show that x3 + x2 + x + 1 is reducible over Q. Does this

Chapter 17, Problem 11E

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QUESTION:

Show that x3 + x2 + x + 1 is reducible over Q. Does this fact contradict the corollary to Theorem 17.4?Reference:

Questions & Answers

QUESTION:

Show that x3 + x2 + x + 1 is reducible over Q. Does this fact contradict the corollary to Theorem 17.4?Reference:

ANSWER:

Step 1 of 3

Consider the function  

Need to show that f(x) is reducible over Q.

Recall the corollary, the factor theorem,

Let F be a field.  And. Then a is a zero of f(x) if and only if is a factor of f(x)

Here,

Since  is a zero of f(x), from the above corollary,  is a factor of f(x).

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