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For the first part of this question:
So if everything was done correctly is -55/16 my answer?
So if everything was done correctly is -55/16 my answer?
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Maybe I'm missing something, but I don't get where the -55/16 comes from. f(x) has no zeroes in the rationals, hence no factors with rational coefficients.
I'm not sure what F5 means, but if it represents the integers modulo 5, both f and g have a common zero, hence a common factor.