a \( \frac{2 x}{(x+2)^{2}} \) b \( \frac{11 x^{2}+14 x+5}{(2 x+1)(x+1)^{2}} \) c \( \frac{x^{2}-2}{x(x-1)^{2}} \) d \( \frac{36 x^{2}+2 x-4}{(2 x-3)(2 x+1)^{2}} \) e \( \frac{3}{(x+2)(x-2)^{2}} \) f \( \frac{3 x+4}{(x+2)(x-1)^{2}} \) 173

Did you know that rational functions like these often show up in calculus when discussing limits, derivatives, and integrals? They help explain how functions behave near their asymptotes or undefined points. So, if you ever find yourself groaning over these fractions, just remember they’re secretly the stars of the calculus show! If you’re looking to simplify your work with these rational functions, pay attention to factoring! A common mistake is to overlook shared factors that can help reduce the fractions. Always check for cancellations before diving into more complicated calculations—it saves time and frustration, plus it’s like finding a hidden shortcut on your math journey!