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This is one of our many quizzes on Rational expressions and equations . Click on this to access more such competitive quizzes on Number System.

Rational expressions and equations Quiz 2

mathematics

 Simplify: $\dfrac{2{{x}^{2}}-17x-9}{2{{x}^{2}}+4x-198}\div \dfrac{4{{x}^{2}}-1}{8{{x}^{2}}+84x-44}$

 Factorize each polynomial and then cancel the terms to reduce to the lowest terms.

  $\text{Given: }\dfrac{2{{x}^{2}}-17x-9}{2{{x}^{2}}+4x-198}\div \dfrac{4{{x}^{2}}-1}{8{{x}^{2}}+84x-44} \\ \Rightarrow \dfrac{2{{x}^{2}}-18x+x-9}{2({{x}^{2}}+11x-9x-99)}\times \dfrac{4(2{{x}^{2}}-x+22x-11)}{{{(2x)}^{2}}-1} \\ \Rightarrow \dfrac{(2x+1)(x-9)}{2(x-9)(x+11)}\times \dfrac{4(x+11)(2x-1)}{(2x+1)(2x-1)} \\ \Rightarrow 2$ 

 Simplify:$\dfrac{2}{5{{x}^{5}}}-\dfrac{3}{{{x}^{2}}}-\dfrac{2-3x}{{{x}^{3}}}$

 Take LCM of the denominator terms and now simplify the numerator by doing the addition/subtraction.

 $\dfrac{2}{5{{x}^{5}}}-\dfrac{3}{{{x}^{2}}}-\dfrac{2-3x}{{{x}^{3}}} \\ \Rightarrow \dfrac{2-15{{x}^{3}}-10{{x}^{2}}+15{{x}^{3}}}{5{{x}^{5}}} \\ \Rightarrow \dfrac{2-10{{x}^{2}}}{5{{x}^{5}}}$

 Simplify: ${{l}^{3}}\left( \dfrac{1}{l-q}-\dfrac{1}{l+q} \right)+{{q}^{3}}\left( \dfrac{1}{q-l}-\dfrac{1}{l+q} \right)$ 

 Solve each bracket separately as much as possible and then take their LCM to simplify further to get the answer.

 ${{l}^{3}}\left( \dfrac{1}{l-q}-\dfrac{1}{l+q} \right)+{{q}^{3}}\left( \dfrac{1}{q-l}-\dfrac{1}{l+q} \right) \\ \Rightarrow \dfrac{2q{{l}^{3}}}{(l-q)(l+q)}+\dfrac{2l{{q}^{3}}}{(l+q)(-(l-q))} \\ \Rightarrow \dfrac{2ql({{l}^{2}}-{{q}^{2}})}{({{l}^{2}}-{{q}^{2}})} \\ \Rightarrow 2ql$

 Solve for $x : \dfrac{\dfrac{1}{x}-\dfrac{1}{x+7}}{\dfrac{14}{{{x}^{2}}+2x-35}} = \dfrac{\dfrac{1}{7-2x}+\dfrac{1}{2x+5}}{\dfrac{48}{4x+35-4{{x}^{2}}}}$.

 Convert the division into a multiplication and then factorize each polynomial. Then, cancel the terms and cross multiply to solve for $x$.

 $\dfrac{\dfrac{1}{x}-\dfrac{1}{x+7}}{\dfrac{14}{{{x}^{2}}+2x-35}}=\dfrac{\dfrac{1}{7-2x}+\dfrac{1}{2x+5}}{\dfrac{48}{4x+35-4{{x}^{2}}}} \\ \Rightarrow \dfrac{x+7-x}{x(x+7)}\times \dfrac{(x+7)(x-5)}{14}=\dfrac{2x+5+7-2x}{(7-2x)(2x+5)}\times \dfrac{(7-2x)(2x+5)}{48} \\ \Rightarrow \dfrac{x-5}{2x}=\dfrac{1}{4} \\ \Rightarrow 2x-10=x\Rightarrow x=10$ 

 Simplify: $\dfrac{\dfrac{2}{{{(s+z)}^{2}}}+\dfrac{1}{(s+z)(s-z)}+\dfrac{2}{{{(s-z)}^{2}}}}{\dfrac{5{{s}^{3}}-3{{z}^{3}}+3{{z}^{2}}s-5{{s}^{2}}z}{{{(s+z)}^{2}}}}$

 Convert the division into a multiplication. Then take LCM and simplify the numerator by doing the addition/subtraction. Then, cancel the terms to get the lowest terms as the required answer.

  $\dfrac{\dfrac{2}{{{(s+z)}^{2}}}+\dfrac{1}{(s+z)(s-z)}+\dfrac{2}{{{(s-z)}^{2}}}}{\dfrac{5{{s}^{3}}-3{{z}^{3}}+3{{z}^{2}}s-5{{s}^{2}}z}{{{(s+z)}^{2}}}} \\ \Rightarrow\dfrac{2{{s}^{2}}+2{{z}^{2}}-4sz+{{s}^{2}}-{{z}^{2}}+2{{s}^{2}}+2{{z}^{2}}+4sz}{{{(s+z)}^{2}}{{(s-z)}^{2}}}\times \dfrac{{{(s+z)}^{2}}}{5{{s}^{3}}-5{{s}^{2}}z+3{{z}^{2}}s-3{{z}^{3}}} \\ \Rightarrow \dfrac{5{{s}^{2}}+3{{z}^{2}}}{{{(s-z)}^{2}}}\times \dfrac{1}{(5{{s}^{2}}+3{{z}^{2}})(s-z)} \\ \Rightarrow \dfrac{1}{{{(s-z)}^{3}}}$ 