EDGE Math 8

Example

Perform the indicated operations. Then simplify your answer.

1. $\frac{x^{2}}{x + 2} - \frac{8 + 2 x}{x + 2} + \frac{2 + x}{x + 2}$

2. $\frac{5 x - 3}{x^{2} - 9} + \frac{12}{x^{2} - 9} - \frac{2 x}{x^{2} - 9}$

Solution

\frac{x^{2}}{x + 2} - \frac{8 + 2 x}{x + 2} + \frac{2 + x}{x + 2}

= \frac{x^{2} - (8 + 2 x) + (2 + x)}{x + 2}

= \frac{x^{2} - 8 - 2 x + 2 + x}{x + 2}

Combine like terms in the numerator.

\frac{x^{2} - 8 - 2 x + 2 + x}{x + 2} = \frac{x^{2} - x - 6}{x + 2}

Factor the numerator. Then cancel common factors in the numerator and in the denominator.

\frac{x^{2} - x - 6}{x + 2} = \frac{(x - 3) (x + 2)}{x + 2} = x - 3

Therefore, $\frac{x^{2}}{x + 2} - \frac{8 + 2 x}{x + 2} + \frac{2 + x}{x + 2} = x - 3.$

\frac{5 x - 3}{x^{2} - 9} + \frac{12}{x^{2} - 9} - \frac{2 x}{x^{2} - 9}

= \frac{5 x - 3 + 12 - 2 x}{x^{2} - 9}

Combine like terms in the numerator.

\frac{5 x - 3 + 12 - 2 x}{x^{2} - 9} = \frac{3 x + 9}{x^{2} - 9}

Factor the numerator and the dominator. Then cancel common factors.

\frac{3 x + 9}{x^{2} - 9} = \frac{3 (x + 3)}{(x + 3) (x - 3)} = \frac{3}{x - 3}

Therefore, $\frac{5 x - 3}{x^{2} - 9} + \frac{12}{x^{2} - 9} - \frac{2 x}{x^{2} - 9} = \frac{3}{x - 3} .$