To add fractions the fractions must be over a common denominator. First, we need to multiply each fraction by the appropriate form of #1# to put them over a common denominator of:
#x(x - 1)(x - 2)#
#2/x + 2/(x - 1) - 2/(x - 2) =>#
#(((x - 1)(x - 2))/((x - 1)(x - 2)) xx 2/x) + ((x(x - 2))/(x(x - 2)) xx 2/(x - 1)) - ((x(x - 1))/(x(x - 1)) xx 2/(x - 2)) =>#
#(2(x - 1)(x - 2))/(x(x - 1)(x - 2)) + (2x(x - 2))/(x(x - 1)(x - 2)) - (2x(x - 1))/(x(x - 1)(x - 2))#
Next, we can expand the numerators for each fraction:
#((2x - 2)(x - 2))/(x(x - 1)(x - 2)) + (2x^2 - 4x)/(x(x - 1)(x - 2)) - (2x^2 - 2x)/(x(x - 1)(x - 2)) =>#
#(2x^2 - 4x - 2x + 4)/(x(x - 1)(x - 2)) + (2x^2 - 4x)/(x(x - 1)(x - 2)) - (2x^2 - 2x)/(x(x - 1)(x - 2)) =>#
#(2x^2 - 6x + 4)/(x(x - 1)(x - 2)) + (2x^2 - 4x)/(x(x - 1)(x - 2)) - (2x^2 - 2x)/(x(x - 1)(x - 2))#
Then, we can add the numerators over the common denominator:
#((2x^2 - 6x + 4) + (2x^2 - 4x) - (2x^2 - 2x))/(x(x - 1)(x - 2)) =>#
#((2x^2 - 6x + 4 + 2x^2 - 4x - 2x^2 + 2x))/(x(x - 1)(x - 2)) =>#
#((2x^2 + 2x^2 - 2x^2 - 6x - 4x + 2x + 4))/(x(x - 1)(x - 2)) =>#
#((2 + 2 - 2)x^2 + (-6 - 4 + 2)x + 4)/(x(x - 1)(x - 2)) =>#
#(2x^2 + (-8)x + 4)/(x(x - 1)(x - 2)) =>#
#(2x^2 - 8x + 4)/(x(x - 1)(x - 2))#
If you need to simplify the denominator it would be:
#(2x^2 - 8x + 4)/((x^2 - x)(x - 2)) =>#
#(2x^2 - 8x + 4)/(x^3 - 2x^2 - x^2 + 2x ) =>#
#(2x^2 - 8x + 4)/(x^3 + (-2 - 1)x^2 + 2x ) =>#
#(2x^2 - 8x + 4)/(x^3 + (-3)x^2 + 2x ) =>#
#(2x^2 - 8x + 4)/(x^3 - 3x^2 + 2x )#
