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Question
Take away:
\[\frac{5 a^2}{2} + \frac{3 a^3}{2} + \frac{a}{3} - \frac{6}{5} \text { from } \frac{1}{3} a^3 - \frac{3}{4} a^2 - \frac{5}{2}\]
Solution
The difference is given by:
\[\left( \frac{1}{3} a^3 - \frac{3 a^2}{4} - \frac{5}{2} \right) - \left( \frac{5 a^2}{2} + \frac{3 a^3}{2} + \frac{a}{3} - \frac{6}{5} \right)\]
\[ = \frac{1}{3} a^3 - \frac{3 a^2}{4} - \frac{5}{2} - \frac{5 a^2}{2} - \frac{3 a^3}{2} - \frac{a}{3} + \frac{6}{5}\]
\[= \frac{1}{3} a^3 - \frac{3 a^3}{2} - \frac{3 a^2}{4} - \frac{5 a^2}{2} - \frac{a}{3} - \frac{5}{2} + \frac{6}{5}\] (Collecting like terms)
= \[\left( \frac{2 - 9}{6} \right) a^3 + \left( \frac{- 3 - 10}{4} \right) a^2 - \frac{a}{3} + \left( \frac{- 25 + 12}{10} \right)\]
\[= - \frac{7}{6} a^3 - \frac{13}{4} a^2 - \frac{a}{3} - \frac{13}{10}\] (Combining like terms)
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