Question

How do you simplify `(x-2)/(x^3-8)`?

Answer 1

`1/(x^2+2x+4)`

Explanation:

`x^3-8 = (x-2)(x^2+2x+4)`
The above uses the rule `x^3-y^3 = (x-y)(x^2+xy+y^2)`

`(x-2)/(x^3-8) = (x-2)/((x-2)(x^2+2x+4)) = 1/(x^2+2x+4)`

Answer 2

`1/(x^2+2x+4),x!=2`

Explanation:

`" "(x-2)/(x^3-8)`

In rational expressions, any variable on the denominator must not equal zero, so we can find restrictions from the original equation initially to save time in the end:
`" "x^3-8!=0`
`" "x^3!=8`
`" "x!=2`

Continue solving by putting `8` into exponent form.
`=(x-2)/(x^3-2^3)`

Factor the difference of cubes (`a^3-b^3=(a-b)(a^2+ab+b^2)`).
`=(x-2)/((x-2)(x^2+2x+4))`

Reduce the fraction with `x-2`.
`=cancel(x-2)/(cancel(x-2)(x^2+2x+4))`
`=1/(x^2+2x+4),x!=2`