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if-f-x-2x-3-x-2-1-x-2-1-Find-the-value-for-which-f-x-is-undefied-2-Express-f-x-in-partial-fraction-




Question Number 27578 by Mcfaithal last updated on 10/Jan/18
if f(x)=((2x−3)/((x^2 −1)(x+2)))  1. Find the value for which f(x) is undefied.  2.Express f(x) in partial fraction
$$\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2x}−\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}+\mathrm{2}\right)} \\ $$$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{for}\:\mathrm{which}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{undefied}. \\ $$$$\mathrm{2}.\mathrm{Express}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{in}\:\mathrm{partial}\:\mathrm{fraction} \\ $$
Commented by abdo imad last updated on 10/Jan/18
D=R−{1,−1,−2}  f(x)= (a/(x−1)) + (b/(x+1))  +(c/(x+2))  we have f(x)=  ((2x−3)/((x−1)(x+1)(x+2)))  ⇒ a=  ((−1)/6)   ,  b= (5/2)    , c= ((−7)/3)and  f(x)=  ((−1)/(6(x−1)))  + (5/(2(x+1))) − (7/(3(x+2)))  .
$${D}={R}−\left\{\mathrm{1},−\mathrm{1},−\mathrm{2}\right\} \\ $$$${f}\left({x}\right)=\:\frac{{a}}{{x}−\mathrm{1}}\:+\:\frac{{b}}{{x}+\mathrm{1}}\:\:+\frac{{c}}{{x}+\mathrm{2}} \\ $$$${we}\:{have}\:{f}\left({x}\right)=\:\:\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)} \\ $$$$\Rightarrow\:{a}=\:\:\frac{−\mathrm{1}}{\mathrm{6}}\:\:\:,\:\:{b}=\:\frac{\mathrm{5}}{\mathrm{2}}\:\:\:\:,\:{c}=\:\frac{−\mathrm{7}}{\mathrm{3}}{and} \\ $$$${f}\left({x}\right)=\:\:\frac{−\mathrm{1}}{\mathrm{6}\left({x}−\mathrm{1}\right)}\:\:+\:\frac{\mathrm{5}}{\mathrm{2}\left({x}+\mathrm{1}\right)}\:−\:\frac{\mathrm{7}}{\mathrm{3}\left({x}+\mathrm{2}\right)}\:\:. \\ $$

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