MHB Simplifying polynomial fraction

[coolbeans33]

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

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

to e^x(1-x)²/(1+x²)²

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

 

[Chris L T521]

Re: simplifying polynomial fraction

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

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

to e^x(1-x)²/(1+x²)²

I understand the first step, but don't get how they got to e^x(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?