Simplify the following: \[- \frac{1}{2} a^2 b^2 c + \frac{1}{3}a b^2 c - \frac{1}{4}ab c^2 - \frac{1}{5}c b^2 a^2 + \frac{1}{6}c b^2 a - \frac{1}{7} c^2 ab + \frac{1}{8}c a^2 b .\]
Solution
\[- \frac{1}{2} a^2 b^2 c + \frac{1}{3}a b^2 c - \frac{1}{4}ab c^2 - \frac{1}{5}c b^2 a^2 + \frac{1}{6}c b^2 a - \frac{1}{7} c^2 ab + \frac{1}{8}c a^2 b\]
\[= - \frac{1}{2} a^2 b^2 c - \frac{1}{5}c b^2 a^2 + \frac{1}{3}a b^2 c + \frac{1}{6}c b^2 a - \frac{1}{4}ab c^2 - \frac{1}{7} c^2 ab + \frac{1}{8}c a^2 b\] (Collecting like terms)
= \[\left( \frac{- 5 - 2}{10} \right) a^2 b^2 c + \left( \frac{2 + 1}{6} \right)c b^2 a^2 + \left( \frac{- 7 - 4}{28} \right) c^2 ab + \frac{1}{8}c a^2 b\]
\[= - \frac{7}{10} a^2 b^2 c + \frac{1}{2}a b^2 c - \frac{11}{28}ab c^2 + \frac{1}{8} a^2 bc\] (Combining like terms)
