Question

1. \\(\frac{x}{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}\\)
2. \\(\frac{ab^3 - 9ab}{12ab^3 + 12ab - 144a}\\)

Explanation:

Step1: Factor numerator and denominator

Factor $y^2-5y-24$ into $(y-8)(y+3)$; factor $y^2+3y$ into $y(y+3)$.
Expression: $\frac{(y-8)(y+3)}{y(y+3)}$

Step2: Cancel common factors

Cancel $(y+3)$ from numerator and denominator (where $y
eq -3$).
Expression: $\frac{y-8}{y}$

Step3: Find domain restrictions

Denominator cannot be 0: $y(y+3)
eq 0 \implies y
eq 0, y
eq -3$.

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Step1: Factor numerator and denominator

Factor $ab^3-9ab$ into $ab(b^2-9)=ab(b-3)(b+3)$; factor $12ab^2+12ab-144a$ into $12a(b^2+b-12)=12a(b+4)(b-3)$.
Expression: $\frac{ab(b-3)(b+3)}{12a(b+4)(b-3)}$

Step2: Cancel common factors

Cancel $a$ and $(b-3)$ from numerator and denominator (where $a
eq 0, b
eq 3$).
Expression: $\frac{b(b+3)}{12(b+4)}$

Step3: Find domain restrictions

Denominator cannot be 0: $12a(b+4)(b-3)
eq 0 \implies a
eq 0, b
eq 3, b
eq -4$.

Answer:

1. Simplified form: $\frac{y-8}{y}$; Domain: All real numbers except $y=0$ and $y=-3$
2. Simplified form: $\frac{b(b+3)}{12(b+4)}$; Domain: All real numbers except $a=0$, $b=3$, and $b=-4$