SOLUTION: The polynomial {{{x^8 - 1}}} is factored as {{{x^8 - 1 = p_1(x) * p_2(x)}}}....{{{p_k(x)}}},where each factor {{{p_i(x)}}} is a non-constant polynomial with real coefficients. Fin

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Question 1157739: The polynomial is factored as
....,where each factor is a non-constant polynomial with real coefficients. Find the largest possible value of k.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!



=
=
=

The factorization can be done in a different order; but the final factorization will be the same.

Neither x^4+1 nor x^2+1 can be factored into the product of non-constant polynomials with real coefficients.

ANSWER: 4
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.

     =  =  = .     (1)



The factor    also can be factored over real numbers


     =  -  =  -  = .


   +--------------------------------------------------------------------------------------------------------+
   |  Far not everyone knows about this tricky decomposition; but those who are trained in Math, know it.   |
   |      See the lesson  Advanced factoring  in this site.                                                 |
   +--------------------------------------------------------------------------------------------------------+



Therefore, decomposition (1) can be continue farther


     =  =  = .



Three remaining quadratic polynomials CAN NOT be factored further over real numbers.



Therefore,  k = 5  is the largest value of " k "  under the problem's question.      ANSWER

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Solved.

The answer and the statement  " k = 4 "  by  @greenestamps  is not correct.



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