Viète's Formulas/Examples/Quartic

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Example of Use of Viète's Formulas

Consider the quartic equation:

$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$

Let its roots be denoted $x_1$, $x_2$, $x_3$ and $x_4$.


Then:

\(\ds x_1 + x_2 + x_3 + x_4\) \(=\) \(\ds -a_1\)
\(\ds x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_1 + x_1 x_3 + x_2 x_4\) \(=\) \(\ds a_2\)
\(\ds x_1 x_2 x_3 + x_2 x_3 x_4 + x_1 x_2 x_4 + x_1 x_3 x_4\) \(=\) \(\ds -a_3\)
\(\ds x_1 x_2 x_3 x_4\) \(=\) \(\ds a_4\)


Proof

A specific instance of Viète's Formulas for $n = 4$.

$\blacksquare$


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