Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$.  What is the largest of the three numbers?  Express your answer in simplest radical form.
I notice that the three equations involve the coefficients of a cubic polynomial with roots $a,b,$ and $c$. That is, if I let $p(x)=(x-a)(x-b)(x-c)$, then I can write $p(x)=x^3-2x^2-7x+14$. To find the largest root of $p(x)$, I could try to factor it or use the rational root theorem. Factoring by grouping, I see that $p(x)=x^3-7x-2x^2+14=x(x^2-7)-2(x^2-7)=(x-2)(x^2-7)$. So the roots of $p(x)$ are $x=2$ and $x=\pm\sqrt{7}$. The largest of these roots is $x=\sqrt{7}$. Therefore, the largest of the three numbers $a,b,$ and $c$ is $\sqrt{7}$. 