# How do you combine  [(3x - 5)/(3 - x)] + [(1)/(1 - x)] - [(x^2 - 1)/(x^2 - 4x + 3)]?

Jun 5, 2015

You must factorize the third denominator and then sum the fractions with the LCM algorythm.

First we factorize ${x}^{2} - 4 x + 3 = \left(x - 1\right) \left(x - 3\right)$
Now, a good way to simplify the operation, let's change the sign to the first two fractions so their denominators are divisors of the third one. We've got:

(-3x+5)/(x-3)-1/(x-1)-(x^2-1)/((x-1)(x-3)

The Least Common Multiple of the three denominators is the third one, which will be the denominator of the sum, now, follows dividing by each denominator and multiply by its numerator to get each term of the sum.

$\frac{\left(- 3 x + 5\right) \left(x - 1\right) - \left(x - 3\right) - \left({x}^{2} - 1\right)}{\left(x - 1\right) \left(x - 3\right)} = \frac{- 3 {x}^{2} + 3 x + 5 x - 5 - x + 3 - {x}^{2} + 1}{\left(x - 1\right) \left(x - 3\right)} = \frac{- 4 {x}^{2} + 7 x - 1}{\left(x - 1\right) \left(x - 3\right)}$