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Question

Let g(x) be a polynomial with leading coefficient 1, whose three roots are the reciprocals of the three roots of f(x)=x^3+ax^2+bx+c, where 1<a<b<c. What is g(1) in terms of a,b, and c?

Accepted Answer

Note that f(1/x) has the same roots as g(x), if it is multiplied by some monomial so that the leading term is x^3 they will be equal. We have f(1/x) = 1x^3 + ax^2+bx + c so we can see that g(x) = x^3cf(1/x) Therefore g(1) = 1cf(1) = 1+a+b+cc