Identifying the type of expression

AI Thread Summary
The discussion revolves around identifying types of mathematical expressions, specifically focusing on the simplification of two given expressions. The first expression simplifies to a fractional form, while the second results in a quadratic expression, leading to a debate on terminology regarding their classification. Participants note that constant and square terms can cancel in the second expression, indicating a subjective approach to simplification. A follow-up question introduces additional expressions, prompting further inquiry into whether they are considered fractional. The conversation highlights the nuances in classifying mathematical expressions based on their simplified forms.
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TL;DR Summary: ##(1+ \frac1x)^2 - (1-\frac1x)^2##
##(z+2)^2 -5(z+2)##

Upon simplifying the first I get ##\frac4x##. So isn’t the first expression fractional?
Upon simplifying the second I get a Quadratic expression.
 
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1. If you want to call it that, yes. One could also say that constant and square terms cancel.
2. I get a product -- matter of taste which is simpler

##\ ##
 
Thanks man :)
 
What about ##(x^2+3)^{-\frac13} + \frac23 x^2(x^2+3)^{-\frac43}## ? Fractional expression?
or this ##(x^2+3)^{-\frac13} + 2 x^2(x^2+3)^{-\frac43}## ?
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
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