# Exercise ID244

Algebra → Polynomials → Division of polynomial
[Level: ] [Number of helps: 0] [Number of pictures: 0] [Number of steps: 18] [Number of characters: 0]

Simplify the following expression:

$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}$

$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}{+}\frac{1}{1+\frac{b}{a}}{-}\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}={A}:{B}\phantom{\rule{0ex}{0ex}}$ ${A}{=}\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}{+}\frac{1}{1+\frac{b}{a}}{-}\frac{1}{1-\frac{b}{a}}\phantom{\rule{0ex}{0ex}}$ ${A}=\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}$ $=\frac{\frac{{a}^{2}+{b}^{2}}{{a}{b}}}{\frac{{a}^{2}-{b}^{2}}{{a}{b}}}+\frac{\frac{1}{1}}{\frac{a+b}{a}}-\frac{\frac{1}{1}}{\frac{a-b}{a}}$ $=\frac{{a}^{2}+{b}^{2}}{{a}^{2}-{b}^{2}}+\frac{a}{a+b}-\frac{a}{a-b}$ $=\frac{{a}^{2}+{b}^{2}+a\left(a-b\right)-a\left(a+b\right)}{\left(a+b\right)·\left(a-b\right)}$ $=\frac{{a}^{2}+{b}^{2}{+}{{a}}^{2}-ab{-}{{a}}^{2}-ab}{\left(a+b\right)·\left(a-b\right)}$ $=\frac{{a}^{2}+{b}^{2}-2ab}{\left(a+b\right)·\left(a-b\right)}$ $=\frac{{\left(a-b\right)}^{{2}}}{\left(a+b\right)·{\left(}{a}{-}{b}{\right)}}$ $=\frac{a-b}{a+b}$ ${B}{=}\frac{\frac{a+b-a+3b}{a+b}}{\frac{3a+b-3a+3b}{a-b}}\phantom{\rule{0ex}{0ex}}$ ${B}=\frac{\frac{a+b-a+3b}{a+b}}{\frac{3a+b-3a+3b}{a-b}}\phantom{\rule{0ex}{0ex}}$ $=\frac{\frac{{4}{b}}{a+b}}{\frac{{4}{b}}{a-b}}$ $=\frac{a-b}{a+b}$ ${A}:{B}=\frac{a-b}{a+b}:\frac{a-b}{a+b}\phantom{\rule{0ex}{0ex}}$ ${A}:{B}=\frac{a-b}{a+b}:\frac{a-b}{a+b}$ $=\frac{{a}{-}{b}}{{a}{+}{b}}·\frac{{a}{+}{b}}{{a}{-}{b}}$ $=1$
$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}=1$

$\frac{a}{b}{:}\frac{{c}}{{d}}=\frac{a}{b}{·}\frac{{d}}{{c}}$

${a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right)$