Question P.6.9: Adding Rational Expressions with Different Denominators Add:...

Step 1  Find the least common denominator. Start by factoring the denominators.

x² + x – 2 = (x + 2)(x – 1)

x² – 1 = (x + 1)(x – 1)

The factors of the first denominator are x + 2 and x – 1. The only factor from the second denominator that is not listed is x + 1. Thus, the least common denominator is

(x + 2)(x – 1)(x + 1).

Step 2  Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of (x + 2)(x – 1)(x + 1). We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.

Because \frac{x+1}{x+1}=1 and \frac{x+2}{x+2}=1, we are not changing the value of either rational expression, only its appearance.

Now we are ready to perform the indicated addition.

\frac{x + 3}{x² + x – 2} + \frac{2}{x² – 1}
This is the given problem.

= \frac{x + 3}{(x + 2)(x – 1)} + \frac{2}{(x + 1)(x – 1)}
Factor the denominators. The LCD is (x + 2)(x – 1)(x + 1).

= \frac{(x + 3)(x+1)}{(x + 2)(x – 1)(x+1)} + \frac{2(x+2)}{(x+2)(x + 1)(x – 1)}
Rewrite as equivalent expressions with the LCD.

Step 3  Add numerators, putting this sum over the LCD.

=\frac{x² + 4x + 3 + 2x + 4}{(x + 2)(x – 1)(x + 1)}
Perform the multiplications in the numerator.

=\frac{x² + 6x + 7}{(x + 2)(x – 1)(x + 1)}, x ≠ -2, x ≠ 1, x ≠ -1
Combine like terms in the numerator: 4x + 2x = 6x and 3 + 4 = 7.

Step 4  If necessary, simplify. Because the numerator is prime, no further simplification is possible.