The comparing-coefficients idea extends directly to cubics. If \(ax^3+bx^2+cx+d=0\) has roots \(\alpha\), \(\beta\) and \(\gamma\), expanding \(a(x-\alpha)(x-\beta)(x-\gamma)=0\) and matching coefficients gives three symmetric results.The squares identity extends too: \(\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)\).Reversing the comparison, a cubic with roots \(\alpha\), \(\beta\), \(\gamma\) is \(x^3-\left(\sum\alpha\right)x^2+\left(\sum\alpha\beta\right)x-\alpha\beta\gamma=0\).The signs alternate \(-,\ +,\ -\) starting from the \(x^2\) coefficient. Divide through by \(a\) first, then read the symmetric values straight off the coefficients.

The comparing-coefficients idea extends directly to cubics. If \(ax^3+bx^2+cx+d=0\) has roots \(\alpha\), \(\beta\) and \(\gamma\), expanding \(a(x-\alpha)(x-\beta)(x-\gamma)=0\) and matching coefficients gives three symmetric results.

From two roots to three

The squares identity extends too: \(\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)\).

Reversing the comparison, a cubic with roots \(\alpha\), \(\beta\), \(\gamma\) is \(x^3-\left(\sum\alpha\right)x^2+\left(\sum\alpha\beta\right)x-\alpha\beta\gamma=0\).

The signs alternate \(-,\ +,\ -\) starting from the \(x^2\) coefficient. Divide through by \(a\) first, then read the symmetric values straight off the coefficients.

Result	Formula
Sum of roots, \(\sum\alpha\)	\(\alpha+\beta+\gamma=-\dfrac{b}{a}\)
Sum of products of two roots, \(\sum\alpha\beta\)	\(\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a}\)
Product of roots	\(\alpha\beta\gamma=-\dfrac{d}{a}\)

Cubic Equations and Their Roots

Cubic Equations and Their Roots