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Algebra

# Vieta's Formula - Higher Degrees

$$\alpha$$, $$\beta$$, and $$\gamma$$ are the roots of the cubic equation $$x^3-x^2+2x-13=0$$. If $$\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a+b$$?

Given that 2 roots of $$f(x) = x^3 + ax + b$$ are 2 and 7, what is $$a+b$$?

What is the product of all roots to the equation

\begin{align} & (x-1)(x-2)(x-3) + (x-2)(x-3)(x-4) \\ + & (x-3)(x-4)(x-5) + (x-4)(x-5)(x-6) \\ + & (x-5)(x-6)(x-7) + (x-6)(x-7)(x-8) =0 ? \end{align}

Details and assumptions

Clarification: Make sure you scroll to the right (if need be) to see the full equation. This problem ends with a "?".

If the quartic $$x^4 + 3x^3 + 11 x^2 + 15 x + A$$ has roots $$k, l, m$$ and $$n$$ such that $$kl = mn$$, determine $$A$$.

If the cubic equation $$x^3-3x^2+5=0$$ has three roots $$\alpha$$, $$\beta$$ and $$\gamma$$, what is the value of $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)?$

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