Take away:

Solution:

(i) The difference is given by:

$\left(\frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4}\right)-\left(\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x\right)$

$=\frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4}-\frac{6}{5} x^{2}+\frac{4}{5} x^{3}-\frac{5}{6}-\frac{3}{2} x$

$=\frac{x^{3}}{3}+\frac{4}{5} x^{3}-\frac{5}{2} x^{2}-\frac{6}{5} x^{2}+\frac{3}{5} x-\frac{3}{2} x+\frac{1}{4}-\frac{5}{6}$            (Combining 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)

(ii) The difference is given by:

$\left(\frac{1}{3} a^{3}-\frac{3 a^{2}}{4}-\frac{5}{2}\right)-\left(\frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}-\frac{6}{5}\right)$

$=\frac{1}{3} a^{3}-\frac{3 a^{2}}{4}-\frac{5}{2}-\frac{5 a^{2}}{2}-\frac{3 a^{3}}{2}-\frac{a}{3}+\frac{6}{5}$

$=\frac{1}{3} a^{3}-\frac{3 a^{3}}{2}-\frac{3 a^{2}}{4}-\frac{5 a^{2}}{2}-\frac{a}{3}-\frac{5}{2}+\frac{6}{5}$           (Combining 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)

(iii) The difference is given by:

$\left(\frac{7}{2}-\frac{x}{3}-\frac{x^{2}}{5}\right)-\left(\frac{7 x^{3}}{4}+\frac{3 x^{2}}{5}+\frac{x}{2}+\frac{9}{2}\right)$

$=\frac{7}{2}-\frac{x}{3}-\frac{x^{2}}{5}-\frac{7 x^{3}}{4}-\frac{3 x^{2}}{5}-\frac{x}{2}-\frac{9}{2}$

$=\frac{7}{2}-\frac{9}{2}-\frac{x}{3}-\frac{x}{2}-\frac{x^{2}}{5}-\frac{3 x^{2}}{5}-\frac{7 x^{3}}{4}$               (Combining like terms)

$=\left(\frac{7-9}{2}\right)+\left(\frac{-2-3}{6}\right) x+\left(\frac{-1-3}{5}\right) x^{2}-\frac{7 x^{3}}{4}$

 

$=-1-\frac{5 x}{6}-\frac{4 x^{2}}{5}-\frac{7 x^{3}}{4}$               (Combining like terms)

(iv) The difference is given by:

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

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

$=\frac{1}{3}-\frac{1}{2}-\frac{y}{2}-\frac{5}{3} y^{2}-\frac{7 y^{2}}{3}-\frac{y^{3}}{3}$          (Combining 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)

(v) The difference is given by:

$\left(\frac{3}{2} a b-\frac{7}{4} a c-\frac{5}{6} b c\right)-\left(\frac{2}{3} a c-\frac{5}{7} a b+\frac{2}{3} b c\right)$

$=\frac{3}{2} a b-\frac{7}{4} a c-\frac{5}{6} b c-\frac{2}{3} a c+\frac{5}{7} a b-\frac{2}{3} b c$

$=\frac{3}{2} a b+\frac{5}{7} a b-\frac{7}{4} a c-\frac{2}{3} a c-\frac{5}{6} b c-\frac{2}{3} b c$            (Combining like terms)

$=\left(\frac{21+10}{14}\right) a b+\left(\frac{-21-8}{12}\right) a c+\left(\frac{-5-4}{6}\right) b c$

$=\frac{31}{14} a b-\frac{29}{12} a c-\frac{3}{2} b c$                          (Combining like terms)