simplify: 2x/(x-1) + 5/x - 3x/(x+1)

Expert Answers
hala718 eNotes educator| Certified Educator

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

First we will find the common denomenator:

x(x-1)(x+1)= x(x^2 -1)

==>[ 2x(x)(x+1) + 5(x^1 -1) -3x(x)(x-1)]/x(x^2 -1)

=  [2x^3 + 2x^2 + 5x^2 -5 -3x^3 + 3x^2]/x(x^2 -1)

= [-x^3 + 10x^2  -5]/(x^3 -x)

giorgiana1976 | Student

In order to calculate the sum or difference of 3 quotients, we have to check if they have a common denominator,which is not the case.

To calculate the least common denominator (LCD), we'll multiply all 3 denominators of the ratios:

(x-1)(x)(x+1)

To calculate the expression we'll do the steps:

- we'll multiply 2x/(x-1) by x*(x+1)

- we'll multiply 5/x by (x^2-1)

-  we'll multiply 3x/(x+1) by x*(x-1)

We'll get:

[2x^2*(x+1) + 5(x^2-1) + 3x^2*(x-1)]/(x-1)(x)(x+1)

We'll open the brackets:

2x^3 + 2x^2 + 5x^2 - 5 + 3x^3 - 3x^2

We'll group like terms and we'll factorize and we'll get:

(5x^3+4x^2-5)/(x-1)(x)(x+1)

neela | Student

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

Solution:

 

The LCM of the denominators is (x-1)x(x+1).

There are 3 terms in the given expression. The equivalent rational expressions of each term when the denominator is LCM or (x-1)x(x+1) is as follows:

2x/(x-1) = 2x*x(x+1) /[(x-1)x(x+1)]

5/x = 5(x-1)(x+1)/[(x-1)x(x+1)]

-3x/(x+1) = 3x(*x(x-1)/(x-1)x(x+1)

Adding the numerators as the denominator is common, we get:

Nr : 2x*x(x+1)+5(x-1)(x+1) -3x(x-1)x =

2x^3+2x^2 + 5x^2-5  -3x^3+3x^2

= -x^3 +10x^2 -5= = -(x^3-10x^2+5)

Therefore the given expression = -(x^3-10x^2+5)/[(x-1)x(x+1)]

= -(x^3-10x^2+5)/[x(x^2-1)]

-(x^3-10x^2+5)(x^3-x)

 

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