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Question Number 69662 by aliesam last updated on 26/Sep/19
prove that    ((2x^3 −x^2 −2x+1)/(x^3 +1)) + ((x^3 +1)/(x^4 −2x^3 +3x^2 −2x+1)) = 2
$${prove}\:{that} \\ $$$$ \\ $$$$\frac{\mathrm{2}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}}{{x}^{\mathrm{3}} +\mathrm{1}}\:+\:\frac{{x}^{\mathrm{3}} +\mathrm{1}}{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}}\:=\:\mathrm{2} \\ $$
Answered by MJS last updated on 26/Sep/19
(((x−1)(2x−1)(x+1))/((x+1)(x^2 −x+1)))+(((x+1)(x^2 −x+1))/((x^2 −x+1)^2 ))=  =(((x−1)(2x−1)+(x+1))/(x^2 −x+1))=((2x^2 −2x+2)/(x^2 −x+1))=2
$$\frac{\left({x}−\mathrm{1}\right)\left(\mathrm{2}{x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)}{\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)}+\frac{\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }= \\ $$$$=\frac{\left({x}−\mathrm{1}\right)\left(\mathrm{2}{x}−\mathrm{1}\right)+\left({x}+\mathrm{1}\right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}=\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}=\mathrm{2} \\ $$
Commented by aliesam last updated on 26/Sep/19
thank you sir
$${thank}\:{you}\:{sir}\: \\ $$

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