Respuesta :
Answer:
[tex] e^x (x- \frac{\sqrt{2}}{2}) (x+\frac{\sqrt{2}}{2})[/tex]
Step-by-step explanation:
For this case we have the following expression:
[tex] x^2 e^x - \frac{1}{2}e^x[/tex]
And we want to factorize this, the first step on this case would be taking common factor [tex] e^x[/tex] and we got this:
[tex] e^x (x^2 -\frac{1}{2})[/tex]
Now we can apply this case of factorization called difference of perfect squares:
[tex] a^2 -b^2 = (a-b)(a+b)[/tex]
For this case [tex] a = x , b = \frac{1}{\sqrt{2}}[/tex]
And if we apply this we got:
[tex] e^x (x- \frac{1}{\sqrt{2}})(1+ \frac{1}{\sqrt{2}})[/tex]
Now we can rationalize the expression with the square root on the denominator like this:
[tex] \frac{1}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}[/tex]
And if we replace this we got:
[tex] e^x (x- \frac{\sqrt{2}}{2}) (x+\frac{\sqrt{2}}{2})[/tex]
And that would be our final expression.
