Question

In Exercise,factor the expression.
x^2e^x - 1/2e^x

Answer 1

`e^x (x- (√(2))/(2)) (x+(√(2))/(2))`

Explanation:

For this case we have the following expression:

`x^2 e^x - (1)/(2)e^x`

And we want to factorize this, the first step on this case would be taking common factor
`e^x` and we got this:

`e^x (x^2 -(1)/(2))`

Now we can apply this case of factorization called difference of perfect squares:

`a^2 -b^2 = (a-b)(a+b)`

For this case
`a = x , b = (1)/(√(2))`

And if we apply this we got:

`e^x (x- (1)/(√(2)))(1+ (1)/(√(2)))`

Now we can rationalize the expression with the square root on the denominator like this:

`(1)/(√(2)) * (√(2))/(√(2)) = (√(2))/(2)`

And if we replace this we got:

`e^x (x- (√(2))/(2)) (x+(√(2))/(2))`

And that would be our final expression.