Question

Take away: 

\[\frac{y^3}{3} + \frac{7}{3} y^2 + \frac{1}{2}y + \frac{1}{2} \text { from } \frac{1}{3} - \frac{5}{3} y^2\]

Answer 1

The difference is given by:

\[\left( \frac{1}{3} - \frac{5}{3} y^2 \right) - \left( \frac{y^3}{3} + \frac{7 y^2}{3} + \frac{y}{2} + \frac{1}{2} \right)\]

\[ = \frac{1}{3} - \frac{5}{3} y^2 - \frac{y^3}{3} - \frac{7 y^2}{3} - \frac{y}{2} - \frac{1}{2}\]

\[= \frac{1}{3} - \frac{1}{2} - \frac{y}{2} - \frac{5}{3} y^2 - \frac{7 y^2}{3} - \frac{y^3}{3}\]   ( Collecting like terms)

= \[\left( \frac{2 - 3}{6} \right) - \frac{y}{2} + \left( \frac{- 5 - 7}{3} \right) y^2 - \frac{y^3}{3}\] 

\[= - \frac{1}{6} - \frac{y}{2} - 4 y^2 - \frac{y^3}{3}\]  (Combining like terms. )