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Full Answer. A double root can be confirmed mathematically by examining the equation for solving a second-degree polynomial. If the discriminant, or square root of b^2-4*a*c, is equal to zero, then the equation has a double root. Finishing calculating the solution of the equation will yield two answers of the exact same magnitude.
This function has a hole at [math]x=2[/math] because [math](x-2)[/math] can be crossed out of both the numerator and the denominator. Since [math](x-1)[/math] is a factor of the denominator, [math]x=1[/math] is a vertical asymptote.