$\left[\begin{array}{l}\text { Vieta's Theorem } \\ \text { [Cubic Polynomials }]\end{array}\right]$

Find the relationship between the coefficients of the cubic equation $a x^{3}+b x^{2}+c x+d=0$, given that the sum of two of its roots is equal to the product of these roots.
Let $u, v, w$ be the roots of our equation and $u+v=uv$. Then $c+d=a(uv+uw+vw-uvw)=auv$, $b+c+d=-a(u+v+w)+auv=-aw$, so $(c+d)(b+c+d)=-a^2uvw=ad$.

## Answer

$(c+d)(b+c+d)=ad$.