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Addition of Rational Expressions

Simplify 2x...

Question

Simplify $(\frac{2 x - 1}{x - 1} - \frac{x + 1}{2 x + 1}) + \frac{x + 2}{x + 1}$ as a quotient of two polynomials in the simplest form.

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Solution

Now, $(\frac{2 x - 1}{x - 1} - \frac{x + 1}{2 x - 1}) + \frac{x + 2}{x + 1}$
$= [\frac{(2 x - 1) (2 x + 1) - (x + 1) (x - 1)}{(x - 1) (2 x + 1)}] + \frac{x + 2}{x + 1}$

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

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

$= \frac{5 x^{3} + 6 x^{2} - 3 x - 2}{2 x^{3} + x^{2} - 2 x - 1}$

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