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Let \(x^8 + 98x^4 + 1 = p(x) q(x),\) where \(p(x)\) and \(q(x)\) are monic, non-constant polynomials with integer coefficients. Find \(p(1) + q(1)\).

 Sep 3, 2022
 #1
avatar+39 
0

Based on the polynomial given, we can assume that the degrees of \(p(x)\) and \(q(x)\) is \(4\).

 

And since we have no other terms like \(x^2\) or \(x^6\), we can assume that  \(p(x)\) is in the form \(x^4+A\)  for an integer \(A\), and that \(q(x)\) is in the form \(x^4+B\)  for any integer \(B\).

 

Thus, expanding, we get \(x^8+(A+B)x^4+AB\). From this, we know that \(A+B=98\) and \(AB=1\).

 

Fortunately, we don't need to find \(A\) and \(B\). What we are trying to find is \(2x^4+(A+B)\), which, after we plug in things, we get \(2+98=\boxed{100}\), which is a perfect, round, number to end the problem.

 Sep 3, 2022
 #2
avatar+33151 
+3

I get the following:

 

 

(Note to billyzhang:  The question specifies integer coefficients, which your A and B are not.)

 Sep 4, 2022

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