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Vieta's Formula

Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.

Higher Degree Polynomials

\(\alpha\), \(\beta\), 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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