Question

simplifying rational expressions
simplify the rational expression.
\\(\frac{x^2 - 16}{x^2 + 6x + 8}\\)
options: \\(\frac{x + 4}{x + 2}\\), \\(\frac{x - 4}{x + 4}\\), \\(\frac{x - 4}{x + 2}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 16\) as a difference of squares: \(x^2 - 16=(x + 4)(x - 4)\).
Factor \(x^2+6x + 8\) by finding two numbers that multiply to 8 and add to 6: \(x^2+6x + 8=(x + 2)(x + 4)\).
So the expression becomes \(\frac{(x + 4)(x - 4)}{(x + 2)(x + 4)}\).

Step2: Cancel common factors

Cancel the common factor \((x + 4)\) (assuming \(x
eq - 4\) to avoid division by zero):
\(\frac{(x + 4)(x - 4)}{(x + 2)(x + 4)}=\frac{x - 4}{x + 2}\).

Answer:

The Right Graph (corresponding to \(\frac{x - 4}{x + 2}\))