fractions

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January 16th 2012, 05:13 PM
Veronica1999

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fractions

Could I get some hints for the attached problem?

Thanks.
January 17th 2012, 12:26 AM
princeps

Re: fractions

If we make substitution : $x=2009$ we can write :

$\frac {(x+3)(x+1)(x-1)(x-3)-9}{(x+5)(x+1)(x-1)(x-5)+135}=\frac {x^4-10x^2}{x^4-26x^2+160}=\frac {x^4-10x^2}{x^4-16x^2-10x^2+160}=$

$=\frac{x^2(x^2-10)}{x^2(x^2-16)-10(x^2-16)}=\frac{x^2}{x^2-16}=\frac {2009^2}{2009^2-4^2}=\frac{2009^2}{2005 \cdot 2013}$

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