SOLUTION: Can these be factored using the methods (GCF, squares, cubes) a4 + 1 a4 - 64

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Question 1184738: Can these be factored using the methods (GCF, squares, cubes)
a4 + 1
a4 - 64
Answer by ikleyn(52777) About Me  (Show Source):
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Can these be factored using the methods (GCF, squares, cubes)
(a) a4 + 1
(b) a4 - 64
~~~~~~~~~~~~~~~ 

(b)  a^4 - 64  can be factored this classic way

         a%5E4-64 = %28a%5E2-8%29%2A%28a%5E2%2B8%29

    over integer numbers.


    It can be factored further as

         a%5E4-64 = %28a%5E2-8%29%2A%28a%5E2%2B8%29 = %28a-sqrt%288%29%29%2A%28a%2Bsqrt%288%29%29%2A%28a%5E2%2B8%29 = %28a-2sqrt%282%29%29%2A%28a%2B2sqrt%282%29%29%2A%28a%5E2%2B8%29

    but just over real numbers, not over integer or rational numbers.


    The factor a%5E2%2B8  can be factored further over COMPLEX numbers.




(a)  a%5E4+%2B+1  can be factored in this way

         a%5E4%2B1 = %28a%5E4+%2B+2a%5E2+%2B+1%29 - 2a%5E2 = %28a%5E2%2B1%29%5E2 - %28sqrt%282%29%2Aa%29%5E2 = 

               = %28a%5E2-sqrt%282%29%2Aa+%2B1%29%2A%28a%5E2%2Bsqrt%282%29%2Aa%2B1%29.


     This factoring works over real numbers and includes the irrational number sqrt%282%29.

     It does not work over integer or rational numbers.

Solved, answered and explained.

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See the lesson
    - Advanced factoring
in this site.