Vieta's Formula for Cubic and Higher-Degree Equations

Cubic case. Consider $ax^3+bx^2+cx+d=0$. By the Fundamental Theorem of Algebra it has exactly three roots $\alpha,\beta,\gamma$ (with multiplicity), so

Comparing coefficients (valid since $a\ne0$):

For a monic cubic ($a=1$) this reads especially cleanly:

General degree $n>3$. The same pattern continues for a monic degree-$n$ equation with roots $\alpha_1,\ldots,\alpha_n$: writing $\sum\alpha_1$ for the sum of all the roots, $\sum\alpha_1\alpha_2$ for the sum of all pairwise products, $\sum\alpha_1\alpha_2\alpha_3$ for the sum of all products taken three at a time, and so on (these are the elementary symmetric sums),

For 4 roots $\alpha,\beta,\gamma,\delta$ the notation simplifies to $\sum\alpha=\alpha+\beta+\gamma+\delta$, $\sum\alpha\beta=\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta$ (all $\binom42=6$ pairs), $\sum\alpha\beta\gamma$ (all $\binom43=4$ triples), and $\sum\alpha\beta\gamma\delta=\alpha\beta\gamma\delta$ (the single product of all four).