Determining zeroes, vertical asymptotes from function

Question

Determine x intercepts, Vertical asymptotes and points of discontinuity from this equation,

 

(x^2-2x-1)/ (x^3-27)

 

The answers are (1 +/- square root of 2) x intercept

 (3) vertical asymptotes

and there is no point of discontinuity

 

 

How do we get these answers?!?!

Thankyou

Answer 1

To get the intercepts you must find the values of x where x² -2x - 1 = 0 (i.e. the numerator is zero).

 

The denominator is the difference of two cubes so it can be factored: (x - 3)(x² +3x +9). There is only one real number where this (the denominator) is zero so there is one vertical asymptote at x=3; and x=3 is also a point of discontinuity.

 

 

Where did you get your answers?