Q.Solve the cubic equation: $2x^3-x^2-18x+9=0$ if sum of two of its roots vanishes.

If $\alpha,\beta,\gamma$ are the roots of the monic cubic $x^3+px^2+qx+r=0$, then $f(x)\equiv(x-\alpha)(x-\beta)(x-\gamma)$. Expanding the right side and matching coefficients with $x^3+px^2+qx+r$ gives

For the biquadratic $x^4+px^3+qx^2+rx+s=0$ with roots $\alpha,\beta,\gamma,\delta$, expanding $(x-\alpha)(x-\beta)(x-\gamma)(x-\delta)$ in the same way gives

In general, for the monic degree-$n$ equation $x^n+p_1x^{n-1}+p_2x^{n-2}+\cdots+p_n=0$ with roots $\alpha_1,\ldots,\alpha_n$,

These relations let any symmetric function of the roots -- one whose value is unchanged by permuting $\alpha,\beta,\gamma,\ldots$ among themselves -- be evaluated directly from the coefficients, without ever solving the equation. A symmetric function is first rewritten in terms of $S_1,S_2,S_3,\ldots$: for example $\sum\alpha^2=S_1^2-2S_2$, and for a cubic $\displaystyle\sum\frac1{\alpha^2}=\Big(\sum\frac1\alpha\Big)^2-2\sum\frac1{\alpha\beta}=\Big(\frac{S_2}{S_3}\Big)^2-2\frac{S_1}{S_3}=\frac{S_2^2-2S_1S_3}{S_3^2}$.