SOLUTION: factorise x^4 + 4^2023

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Question 1208797: factorise x^4 + 4^2023
Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
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factorise x^4 + 4^2023.
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Let' start writing

   x%5E4+%2B+4%5E2023 = x%5E4+%2B+4%5E3%2A4%5E2020 = x%5E4+%2B+64%2A4%5E2020 = x%5E4+%2B+8%5E2%2A%284%5E1010%29%5E2 = x%5E4+%2B+%288%2A4%5E1010%29%5E2.


It has the form x%5E4+%2B+a%5E2,  where  a = 8%2A4%5E1010.



Now we will factor  the binomial  x%5E4+%2B+a%5E2  by some tricky way


      x%5E4+%2B+a%5E2 = %28x%5E4+%2B+2a%2Ax%5E2+%2B+a%5E2%29+-+2a%2Ax%5E2 = %28x%5E2+%2B+a%29%5E2 - %28sqrt%282a%29%2Ax%29%5E2 = 
    

    = factorize as the difference of squares = %28x%5E2+%2B+a+%2B+sqrt%282a%29x%29%2A%28x%5E2+%2B+a+-+sqrt%282a%29x%29 = %28x%5E2+%2B+sqrt%282a%29x+%2B+a%29%2A%28x%5E2+-+sqrt%282a%29x+%2B+a%29.


Now substitute  a = 8%2A4%5E1010  into this formula.  Notice that  sqrt%282a%29 = sqrt%282%2A8%2A4%5E1010%29 = sqrt%2816%2A4%5E1010%29 = 4%2A4%5E505 = 4%5E506.  
You will get


    x%5E4+%2B+4%5E2023 = %28x%5E2+%2B+4%5E506%2Ax+%2B+8%2A4%5E1010%29%2A%28x%5E2+-+4%5E506%2Ax+%2B+8%2A4%5E1010%29,


or, which is the same,


    x%5E4+%2B+4%5E2023 = %28x%5E2+%2B+4%5E506%2Ax+%2B+2%2A4%5E1011%29%2A%28x%5E2+-+4%5E506%2Ax+%2B+2%2A4%5E1011%29.


It is the desired factorization.

Solved.

In whole,  it looks like a miracle.

I would say more :   not only it looks like a miracle,  it  highlight%28highlight%28IS%29%29  a miracle.