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:
\(\ds \paren {-1} a_2\) | \(=\) | \(\ds z_1 + z_2 + z_3 = 3\) | ||||||||||||
\(\ds \paren {-1}^2 a_{1}\) | \(=\) | \(\ds z_1 z_2 + z_1 z_3 + z_2 z_3 = 3\) | ||||||||||||
\(\ds \paren {-1}^3 a_0\) | \(=\) | \(\ds z_1 z_2 z_3 = 1\) |
$\blacksquare$