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Question
Take away:
\[\frac{6}{5} x^2 - \frac{4}{5} x^3 + \frac{5}{6} + \frac{3}{2}x \text { from }\frac{x^3}{3} - \frac{5}{2} x^2 + \frac{3}{5}x + \frac{1}{4}\]
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Solution
The difference is given by:
\[\left( \frac{x^3}{3} - \frac{5}{2} x^2 + \frac{3}{5}x + \frac{1}{4} \right) - \left( \frac{6}{5} x^2 - \frac{4}{5} x^3 + \frac{5}{6} + \frac{3}{2}x \right)\]
\[ = \frac{x^3}{3} - \frac{5}{2} x^2 + \frac{3}{5}x + \frac{1}{4} - \frac{6}{5} x^2 + \frac{4}{5} x^3 - \frac{5}{6} - \frac{3}{2}x\]
\[= \frac{x^3}{3} + \frac{4}{5} x^3 - \frac{5}{2} x^2 - \frac{6}{5} x^2 + \frac{3}{5}x - \frac{3}{2}x + \frac{1}{4} - \frac{5}{6}\] (Collecting like terms)
\[\left( \frac{5 + 12}{15} \right) x^3 + \left( \frac{- 25 - 12}{10} \right) x^2 + \left( \frac{6 - 15}{10} \right)x + \left( \frac{6 - 20}{24} \right)\]
\[= \frac{17}{15} x^3 - \frac{37}{10} x^2 - \frac{9}{10}x - \frac{7}{12}\] (Combining like terms)
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