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Simplifying polynomial fraction

coolbeans33

New member
Sep 19, 2013
23
so I was reading my textbook and was showing steps on applying the quotient rule to the function: y=ex/(1+x2)

it went from (1+x2)(ex)-(ex)(2x)/(1+x2)2

to ex(1-x)2/(1+x2)2

I understand the first step, but don't get how they got to ex(1-x)2 in the numerator. can someone please explain the steps to me?
 

Chris L T521

Well-known member
Staff member
Jan 26, 2012
995
Re: simplifying polynomial fraction

so I was reading my textbook and was showing steps on applying the quotient rule to the function: y=ex/(1+x2)

it went from (1+x2)(ex)-(ex)(2x)/(1+x2)2

to ex(1-x)2/(1+x2)2

I understand the first step, but don't get how they got to ex(1-x)2 in the numerator. can someone please explain the steps to me?
For starters, I think you meant to say
\[\frac{(1+x^2)(e^x) - (e^x)(2x)}{(1+x^2)^2}.\]
The first thing to note here is that there's a common factor of $e^x$ in the numerator, i.e.
\[\frac{(\color{blue}{1+x^2})(\color{red}{e^x}) - (\color{red}{e^x})(\color{blue}{2x})}{(1+x^2)^2} = \frac{\color{red}{e^x}(\color{blue}{1+x^2}-\color{blue}{2x})}{(1+x^2)^2} = \frac{e^x(x^2-2x+1)}{(1+x^2)^2}.\]
Can you take things from here?