How do you combine (2)/(x) + (2)/(x-1) - (2)/(x-2)?

1 Answer
Edric Y.
Nov 8, 2015

(2x^2-8x-4)/(x^3-3x^2-2x)

Explanation:

This could get messy.
All of the denominators need to be the same, and the only way we can change them is by multiplying the term by a special form of 1.

So, to get them to match we will need all of them to have this in the denominator: (x)(x-1)(x-2).

NOTE: Don't bother to FOIL anything out yet...

The first term will be multiplied by ((x-1)(x-2))/((x-1)(x-2)) to give
(2(x-1)(x-2))/(x(x-1)(x-2))

The second term will be multiplied by ((x)(x-2))/((x)(x-2)) to give
(2(x)(x-2))/(x(x-1)(x-2))

The third term will be multiplied by ((x)(x-1))/((x)(x-1)) to give
(2(x)(x-1))/(x(x-1)(x-2))

Now to put it all together...
(2(x-1)(x-2))/(x(x-1)(x-2)) + (2(x)(x-2))/(x(x-1)(x-2)) -(2(x)(x-1))/(x(x-1)(x-2))

With common denominators they can be simply combined like so...

((2(x-1)(x-2)) + (2(x)(x-2)) -(2(x)(x-1)))/(x(x-1)(x-2))

Now we need to FOIL stuff out...

((2(x^2-3x-2)) + (2(x^2-2x)) -(2(x^2-x)))/(x^3-3x^2-2x)

((2x^2-6x-4) + (2x^2-4x) -(2x^2-2x))/(x^3-3x^2-2x)

(2x^2-8x-4)/(x^3-3x^2-2x)

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