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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.

Level 3

Jason and Thompson were solving the quadratic equation \(x^{2} + bx + c = 0\).

Jason wrote down the wrong value of \(b,\) and found the roots to be 6 and 1. Thompson wrote down the wrong value of \(c,\) and found the roots to be \(-4\) and \(-1.\)

What are the actual roots of the equation?

The non-zero roots of \[ax^2+bx+c=0\] are \(r\) and \(s.\) What are the roots of \[cx^2+bx+a = 0?\]

The roots of the polynomial equation \( x^3 + 2 x^2 + 3 x + 4 = 0 \) are \( \alpha, \beta,\) and \(\gamma\). What is the value of

\[ \frac{ \alpha + \beta} { \gamma} + \frac{ \beta + \gamma} { \alpha } + \frac{ \gamma + \alpha } { \beta } ?\]

If \(\alpha\), \(\beta\), and \(\gamma\) are the roots of \(x^3-x-1=0\), compute:

\[\frac{1-\alpha}{1+\alpha}+\frac{1-\beta}{1+\beta}+\frac{1-\gamma}{1+\gamma}\]

\[ \large (x-7)(x-3)(x+5)(x+1)=1680\]

Find the sum of all \(x\) that satisfy the equation above.

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