Viète's Formulas/Examples/Cubic with Equal Roots

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Example of Use of Viète's Formulas

The coefficient of $x$ in the expansion of cubic $\paren {x - 1}^3$ is $3$.


Proof

The expansion:

$\paren {x - 1}^3 = x^3 + a_2 x^2 + a_1 x + a_0$

Viète's Formulas for $z_1 = z_2 = z_3 = 1$ are:

\(\displaystyle \paren {-1}\, a_2\) \(=\) \(\displaystyle z_1 + z_2 + z_3 = 3\)
\(\displaystyle \paren {-1}^2 a_{1}\) \(=\) \(\displaystyle z_1z_2 + z_1z_3 + z_2z_3 = 3\)
\(\displaystyle \paren {-1}^3 a_0\) \(=\) \(\displaystyle z_1 z_2 z_3 = 1\)

$\blacksquare$