G c d
Find the sum of possible values of if the highest common factor of f and is a linear polynomial. FOR THE PROBLEM WRITING PARTY
The answer is 0.
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1 solution
g(x) = x^2+cx-2 = (x-a)(x-b) = x^2-(a+b)x+ab so (a,b) = (-1,+2) or (+1,-2). and c = -(a+b) = -1 or +1. so g(x) = (x+1)(x-2) or (x-1)(x+2).
c= -1: f(x) = x^3-x^2-x-2 for which +2 is a root but -1 is not : f(x) = x^3-x^2-x-2 = (x-2)(x^2+x+1).
c = +1 : f(x) = x^3+x^2-x+2 for which +1 is not a root but -2 is : f(x) = x^3+x^2-x+2 = (x+2)(x^2-x+1).
So both c = -1 and c = +1 result in linear GCDs for f(x) and g(x) which are (x-2) and (x+2) respectively...