...th>, and <imath>c</imath> be the roots of the polynomial <imath>x^3 + kx + 1</imath>. What is the sum<cmath>a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + ...tbf{(A)}~-k\qquad\textbf{(B)}~-k+1\qquad\textbf{(C)}~1\qquad\textbf{(D)}~k-1\qquad\textbf{(E)}~k</imath> ...

...+ 10</imath> has three roots in the complex plane, where <imath>i = \sqrt{-1}</imath>. What is the area of the triangle formed by these three roots? == Solution 1 == ...

== Solution 1 == <imath>x</imath> cannot be <imath>1</imath> because it makes <imath>xy>z</imath> <imath>\rightarrow</imath> <im ...

==Solution 1== ...\cos(\angle B)</imath>. Then <imath>\cos(\frac{\angle B}{2}) = \sqrt{\frac{1+\frac{7}{18}}{2}} = \sqrt{\frac{25}{36}} = \frac{5}{6}</imath> (positive ro ...