Answer: [tex]x^2+2x+4[/tex]
Step-by-step explanation:
The expression given in the exercise is:
[tex]\frac{x^3-8}{x-2}[/tex]
If you descompose the number 8 into its prime factors, you get that:
[tex]8=2*2*2=2^3[/tex]
Therefore, you can rewrite the numerator of the expression as following:
[tex]=\frac{(x^3-2^3)}{(x-2)}[/tex]
For this exercise you need to remember that for a Difference of cubes:
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]
Then, applying this, you get:
[tex]=\frac{(x-2)(x^2+2x+2^2)}{(x-2)}=\frac{(x-2)(x^2+2x+4)}{(x-2)}[/tex]
Now, it is necessary to remember the following:
[tex]\frac{a}{a}=1[/tex]
Knowing the above, you can say that:
[tex]\frac{(x-2)}{(x-2)}=1[/tex]
Therefore applying this, you get that the simplified expression is:
[tex]=x^2+2x+4[/tex]
