Question

Take away:
(i) $\frac{6}{5}{x}^2-\frac{4}{5}{x}^3+\frac{5}{6}+\frac{3}{2}x\mathrm{from}\frac{x^3}{3}-\frac{5}{2}{x}^2+\frac{3}{5}x+\frac{1}{4}$
(ii) $\frac{5a^2}{2}+\frac{3a^3}{2}+\frac{a}{3}-\frac{6}{5}\mathrm{from}\frac{1}{3}{a}^3-\frac{3}{4}{a}^2-\frac{5}{2}$
(iii) $\frac{7}{4}{x}^3+\frac{3}{5}{x}^2+\frac{1}{2}x+\frac{9}{2}\mathrm{from}\frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}$
(iv) $\frac{y^3}{3}+\frac{7}{3}{y}^2+\frac{1}{2}y+\frac{1}{2}\mathrm{from}\frac{1}{3}-\frac{5}{3}{y}^2$
(v) $\frac{2}{3}ac-\frac{5}{7}ab+\frac{2}{3}bc\mathrm{from}\frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc$

Answer 1

(i) The difference is given by:

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

(ii) The difference is given by:

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

(iii) The difference is given by:

$$\begin{gathered} \left(\frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}\right)-\left(\frac{7x^3}{4}+\frac{3x^2}{5}+\frac{x}{2}+\frac{9}{2}\right) \\ =\frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}-\frac{7x^3}{4}-\frac{3x^2}{5}-\frac{x}{2}-\frac{9}{2} \end{gathered}$$
$=\frac{7}{2}-\frac{9}{2}-\frac{x}{3}-\frac{x}{2}-\frac{x^2}{5}-\frac{3x^2}{5}-\frac{7x^3}{4}$ (Collecting like terms)
=$\left(\frac{7-9}{2}\right)+\left(\frac{-2-3}{6}\right)x+\left(\frac{-1-3}{5}\right)x^2-\frac{7x^3}{4}$
$=-1-\frac{5x}{6}-\frac{4x^2}{5}-\frac{7x^3}{4}$ (Combining like terms )

(iv) The difference is given by:

$$\begin{gathered} \left(\frac{1}{3}-\frac{5}{3}{y}^2\right)-\left(\frac{y^3}{3}+\frac{7y^2}{3}+\frac{y}{2}+\frac{1}{2}\right) \\ =\frac{1}{3}-\frac{5}{3}{y}^2-\frac{y^3}{3}-\frac{7y^2}{3}-\frac{y}{2}-\frac{1}{2} \end{gathered}$$
$=\frac{1}{3}-\frac{1}{2}-\frac{y}{2}-\frac{5}{3}{y}^2-\frac{7y^2}{3}-\frac{y^3}{3}$ ( Collecting like terms)
=$\left(\frac{2-3}{6}\right)-\frac{y}{2}+\left(\frac{-5-7}{3}\right)y^2-\frac{y^3}{3}$
$=-\frac{1}{6}-\frac{y}{2}-4y^2-\frac{y^3}{3}$ (Combining like terms. )

(v) The difference is given by:

$$\begin{gathered} \left(\frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc\right)-\left(\frac{2}{3}ac-\frac{5}{7}ab+\frac{2}{3}bc\right) \\ =\frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc-\frac{2}{3}ac+\frac{5}{7}ab-\frac{2}{3}bc \end{gathered}$$

$=\frac{3}{2}ab+\frac{5}{7}ab-\frac{7}{4}ac-\frac{2}{3}ac-\frac{5}{6}bc-\frac{2}{3}bc$ (Collecting like terms )
=$\left(\frac{21+10}{14}\right)ab+\left(\frac{-21-8}{12}\right)ac+\left(\frac{-5-4}{6}\right)bc$
$=\frac{31}{14}ab-\frac{29}{12}ac-\frac{3}{2}bc$ (Combining like terms )