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Let x8+98x4+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+53 
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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 x2 or x6, we can assume that  p(x) is in the form x4+A  for an integer A, and that q(x) is in the form x4+B  for any integer B.

 

Thus, expanding, we get x8+(A+B)x4+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 2x4+(A+B), which, after we plug in things, we get 2+98=100, which is a perfect, round, number to end the problem.

 Sep 3, 2022
 #2
avatar+33654 
+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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