Question

question 5 (multiple choice worth 5 points) (04.01n mc) what is the simplified form of \\(\frac{y^2 + 7y + 12}{y^2 - 2y - 15}\\)? \\(\frac{y - 4}{y - 5}\\) \\(\frac{y - 4}{y + 5}\\) \\(\frac{y + 4}{y - 5}\\) \\(\frac{y + 4}{y + 5}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \( y^2 + 7y + 12 \): find two numbers that multiply to 12 and add to 7, which are 3 and 4. So \( y^2 + 7y + 12=(y + 3)(y + 4) \).
Factor \( y^2 - 2y - 15 \): find two numbers that multiply to -15 and add to -2, which are -5 and 3. So \( y^2 - 2y - 15=(y - 5)(y + 3) \).

Step2: Simplify the fraction

Substitute the factored forms into the original fraction: \( \frac{(y + 3)(y + 4)}{(y - 5)(y + 3)} \).
Cancel out the common factor \( (y + 3) \) (assuming \( y
eq - 3 \) to avoid division by zero), resulting in \( \frac{y + 4}{y - 5} \).

Answer:

\(\frac{y + 4}{y - 5}\) (corresponding to the option with numerator \( y + 4 \) and denominator \( y - 5 \))