Question

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

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 quadratic $y^2-5y-24$.
$y^2-5y-24=(y-8)(y+3)$

Step2: Factor denominator

Factor quadratic $y^2+3y$.
$y^2+3y=y(y+3)$

Step3: Cancel common factors

Eliminate $(y+3)$ (where $y
eq-3$).
$\frac{(y-8)\cancel{(y+3)}}{y\cancel{(y+3)}}=\frac{y-8}{y}$

Step4: Find domain restrictions

Identify values making denominator 0.
$y^2+3y=0 \implies y(y+3)=0 \implies y=0, y=-3$

Answer:

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