Match the equations on the left with the properties on the right.

A) Consider the polynomial $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}$ and use Rolle’s Theorem on [0, 1].

B) Consider the polynomial $\mathrm{P}\left(\mathrm{x}\right)=\frac{1}{3}{\mathrm{ax}}^3+\frac{1}{2}{\mathrm{bx}}^2+\frac{1}{6}\mathrm{cx}$ and use Rolle’s theorem on [0, 1].

C) Let $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ Note that $\mathrm{P}(-1)\mathrm{P}\left(1\right)<0$

$\Rightarrow \mathrm{P}\left(\mathrm{x}\right)$ vanishes at least once in (-1, 1)

D) Let $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ and $\mathrm{Q}\left(\mathrm{x}\right)={\mathrm{x}}^2+\frac{\mathrm{b}}{\mathrm{a}}\mathrm{x}+\frac{\mathrm{c}}{\mathrm{a}}$

$\text{At }1+\frac{\left|\mathrm{b}\right|}{\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}}<0\text{, we get }\mathrm{Q}(-1)\text{ or }\mathrm{Q}\left(1\right)0$

$\text{Since }\mathrm{Q}\left(\mathrm{x}\right)\to \mathrm{\infty }\text{ as }\mathrm{x}\to \pm \mathrm{\infty }\text{, we get }\mathrm{Q}\left(\mathrm{x}\right)=0$

$\text{or }\mathrm{P}\left(\mathrm{x}\right)=0\text{ has a root in }(-\mathrm{\infty },-1)\text{ or }(1,\mathrm{\infty })$.