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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

Question 1157739: The polynomial x%5E8+-+1 is factored as
x%5E8+-+1+=+p_1%28x%29+%2A+p_2%28x%29....p_k%28x%29,where each factor p_i%28x%29 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) About Me  (Show Source):
You can put this solution on YOUR website!


x%5E8-1
= %28x%5E4%2B1%29%28x%5E4-1%29
= %28x%5E4%2B1%29%28x%5E2%2B1%29%28x%5E2-1%29
= %28x%5E4%2B1%29%28x%5E2%2B1%29%28x%2B1%29%28x-1%29

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(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

    x%5E8-1%29 = %28x%5E4-1%29%2A%28x%5E4%2B1%29 = %28x%5E2-1%29%2A%28x%5E2%2B1%29%2A%28x%5E4%2B1%29 = %28x-1%29%2A%28x%2B1%29%2A%28x%5E2%2B1%29%2A%28x%5E4%2B1%29.     (1)



The factor  x%5E4%2B1  also can be factored over real numbers


    x%5E4+%2B+1 = %28x%5E4+%2B+2x%5E2+%2B1%29 - 2x%5E2 = %28x%5E2%2B1%29%5E2 - %28sqrt%282%29%2Ax%29%5E2 = %28x%5E2+%2B+%28sqrt%282%29%29%2Ax+%2B1%29%2A%28x%5E2+-+%28sqrt%282%29%29%2Ax+%2B1%29.


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   |  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.                                                 |
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Therefore, decomposition (1) can be continue farther


    x%5E8-1%29 = %28x%5E4-1%29%2A%28x%5E4%2B1%29 = %28x%5E2-1%29%2A%28x%5E2%2B1%29%2A%28x%5E4%2B1%29 = .



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.