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The polynomial x^4 - 27x^2 + 121 can be factored in a unique way into a product of two quadratic polynomials with integer coefficients and leading coefficient 1. What is the sum of these two polynomials?

 May 13, 2020
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x^4 - 27x^2 + 121    

 

Let us guess  that  the factorization may be

 

(x^2 + ax + 11)  (x^2 - ax + 11)        expand

 

x^4 - ax^3 + 11x^2  + ax^3 - a^2x^2 + 11ax  + 11x^2  - 11ax + 121   simplify

 

x^4  + 22x^2 - a^2x^2 + 121

 

Note  that, equating coefficients,    (22 - a^2) =  -27     rearrange

 

22 + 27  =  a^2

 

49 = a^2

 

Take both roots

 

7 = a    and  -7   = a

 

So we have

 

( x^2  + 7x  + 11 )  (x^2 - 7x + 11)

 

The sum is

 

2x^2  + 22

 

 

cool cool cool 

 May 13, 2020

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