IP3U2 HOT PN1 Q8

Vieta's Formulae relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète.

For a quadratic equation in the form of \(a{{x}^{2}}+bx+c=0\) with roots \({{x}_{1}}\) and \({{x}_{2}}\), it is satisfied that

Sum of Roots: \({{x}_{1}}+{{x}_{2}}=-\frac{b}{a}\)

Product of Roots: \({{x}_{1}}{{x}_{2}}=\frac{c}{a}\)

Show that for a cubic equation in the form of \(a{{x}^{3}}+b{{x}^{2}}+cx+d=0\) with roots \({{x}_{1}}\), \({{x}_{2}}\) and \({{x}_{3}}\), Vieta’s Formulae are given by

\(\begin{matrix} {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-\frac{b}{a} \\ {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{2}}{{x}_{3}}=\frac{c}{a} \\ {{x}_{1}}{{x}_{2}}{{x}_{3}}=-\frac{\mathrm{d}}{a} \\ \end{matrix}\)