# How do you combine #(1)/(x) - (2)/(x^2 + x) + (3)/(x^3 - X^2)#?

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To combine the given fractions, we need a common denominator. The common denominator for the fractions (1)/(x), (2)/(x^2 + x), and (3)/(x^3 - x^2) is (x)(x^2 + x)(x^3 - x^2).

Multiplying the first fraction by (x^2 + x)(x^3 - x^2)/(x^2 + x)(x^3 - x^2), the second fraction by (x)(x^3 - x^2)/(x)(x^3 - x^2), and the third fraction by (x)(x^2 + x)/(x)(x^2 + x), we get:

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

Simplifying this expression, we obtain:

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

Expanding and combining like terms in the numerator, we have:

(x^5 - x^4 + x^3 + x^4 - 2x^3 + 2x^2 + 3x^3 - 3x^2 + 3x) / [(x)(x^2 + x)(x^3 - x^2)]

Simplifying further, we get:

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

Therefore, the combined expression is (x^5 + 3x^3 - x^2 + 3x) / [(x)(x^2 + x)(x^3 - x^2)].

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To combine the given rational expressions, we need to find a common denominator. In this case, the common denominator would be the least common denominator (LCD) of ( x ), ( x^2 + x ), and ( x^3 - x^2 ).

Factoring each denominator:

- ( x ) is already in its simplest form.
- ( x^2 + x = x(x + 1) )
- ( x^3 - x^2 = x^2(x - 1) )

So, the LCD is ( x^2(x - 1)(x + 1) ).

Now, we rewrite each term with the common denominator:

- ( \frac{1}{x} ) becomes ( \frac{x^2(x - 1)(x + 1)}{x^2(x - 1)(x + 1)} )
- ( \frac{2}{x^2 + x} ) becomes ( \frac{2x^2(x - 1)}{x^2(x - 1)(x + 1)} )
- ( \frac{3}{x^3 - x^2} ) becomes ( \frac{3}{x^2(x - 1)(x + 1)} )

Now, putting them together: [ \frac{x^2(x - 1)(x + 1)}{x^2(x - 1)(x + 1)} - \frac{2x^2(x - 1)}{x^2(x - 1)(x + 1)} + \frac{3}{x^2(x - 1)(x + 1)} ]

Combining the terms under the common denominator: [ \frac{x^2(x - 1)(x + 1) - 2x^2(x - 1) + 3}{x^2(x - 1)(x + 1)} ]

Expanding and simplifying the numerator: [ \frac{x^4 - x^2 + x^2 - 2x^2 + 3}{x^2(x - 1)(x + 1)} ] [ \frac{x^4 - 2x^2 + 3}{x^2(x - 1)(x + 1)} ]

So, the combined expression is: [ \frac{x^4 - 2x^2 + 3}{x^2(x - 1)(x + 1)} ]

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