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Given that

$$f(x) = 5x^2 -3x + 7$$ and

$$f(g(x)) = 40 -11x^2-8x^4$$

find all possible values for the sum of the coefficients in the polynomial function $g(x)$.

Typically when two functions are provided in such an arranged method, it would be common to divide the two to get another function that is multiplied to provide the resulting function, however it to appears not be possible when dealing with composite functions. Moreover, I was under the impression that it would not be possible to have multiple possibilities for coefficients to provide a single resultant polynomial. Thus, if anyone would be able to provide some suggestions as what to do next, you would have my thanks.

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Remember that if $g(x) = ax^n+...+bx^3+cx^2+dx+e$ then $g(1) =a+...+b+c+d$ is the sum of all coefficients.

Let $t= g(1)$, then we have $$5t^2-3t+7=f(t)= f(g(1))= 40-11-8$$

So we get $$ 5t^2-3t-14 =0$$

Now solve this equation and you will get an answer.

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