Answer:
The simplest form is [tex]\frac{x+2}{x^2+2}[/tex]
Explanation:
The given is: [tex]\frac{x^2 + x-2}{x^3-x^2+2x+2}[/tex]
1- For the numerator:
[tex]x^2+x-2[/tex] can be factorized to give (x-1)(x+2)
2- For the denominator:
[tex]x^3-x^2+2x-2[/tex]
We will factor this using grouping
Take [tex]x^2[/tex] as a common factor from the first two terms and [tex]2[/tex] as a common factor from the last two terms
This will give us:
[tex]x^3-x^2+2x-2[/tex] [tex]=x^2(x-1) + 2(x-1)[/tex]
Now, take (x-1) as a common factor:
[tex]x^3-x^2+2x-2[/tex] = (x-1)(x²+2)
3- Finally, getting the simplest form:
[tex]\frac{x^2+x-2}{x^3-x^2+2x-2}=\frac{(x-1)(x+2)}{(x-1)(x^2+2)}=\frac{x+2}{x^2+2}[/tex]
Hope this helps :)
