Polynomial transformations and Vieta's formulas

If $\alpha$ is a root of $f(x)$ then $\alpha - 1$ is a root of $f(x+1)$ (and vice-versa). So the sum of roots of $f(x+1)$ is the sum of roots of $f(x)$, minus $3$ which by Vieta' formula is $-a$ minus $3 =-a-3$.

Now, the roots of $f(\frac 1x) = \frac{-2x^3+bx^2+ax+1}{x^3}$ are also the roots of $-2x^3+bx^2+ax+1$, the reciprocal polynomial of $f$. The sum of roots of this polynomial is $\frac b2$ by Vieta's formula.

Finally, $-a-3= \frac b2$. Conclude.

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Polynomials

Algebra Precalculus

Contest Math

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