Question

1. $\frac{x^2 + 3x - 10}{x^2 + 3x}$

what is the simplified form of each rational expression? what is the domain? see example 2.

1. $\frac{y^2 - 5y - 24}{y^2 + 3y}$

Explanation:

Step1: Factor numerator

Factor $y^2-5y-24$: find two numbers that multiply to $-24$ and add to $-5$, which are $-8$ and $3$.
$y^2-5y-24=(y-8)(y+3)$

Step2: Factor denominator

Factor $y^2+3y$ by factoring out the common term $y$.
$y^2+3y=y(y+3)$

Step3: Cancel common factors

Cancel the common factor $(y+3)$ from numerator and denominator (valid when $y+3
eq0$).
$\frac{(y-8)(y+3)}{y(y+3)}=\frac{y-8}{y}$

Step4: Find domain restrictions

The original expression is undefined when denominator is 0:

1. $y(y+3)=0$ so $y=0$ or $y=-3$.

Domain is all real numbers except $y=0$ and $y=-3$.

Answer:

Simplified form: $\frac{y-8}{y}$
Domain: All real numbers except $y = -3$ and $y = 0$ (or written as $\mathbb{R} \setminus \{-3, 0\}$)