Answer
$\dfrac{4(y - 3)}{y(y + 5)}$
Restrictions: $y \ne -5, 0, 2$
Work Step by Step
Factor terms to simplest forms:
$\dfrac{(y - 3)(y - 2)}{y^3} \div \dfrac{(y + 5)(y - 2)}{4y^2}$
To divide one rational expression by another, multiply the first expression by the reciprocal of the second expression:
$=\dfrac{(y - 3)(y - 2)}{y^3} \cdot \dfrac{4y^2}{(y + 5)(y - 2)}$
Rewrite as one expression:
$=\dfrac{(y - 3)(y - 2) \cdot 4y^2}{y^3 \cdot (y + 5)(y - 2)}$
Factor out common terms in the numerator and denominator:
$=\dfrac{4(y - 3)}{y(y + 5)}$
Restrictions occur when the denominator is $0$, meaning the expression becomes undefined. To find the restrictions, set each factor in the denominators of all rational expressions equal to zero and solve:
First factor:
$y^3 = 0$
Take the cube root:
$y = 0$
Second factor:
$4y^2 = 0$
Divide each side of the equation by $4$:
$y^2 = 0$
Take the square root:
$y = 0$
Third factor:
$y + 5 = 0$
Subtract $5$ from each side of the equation:
$y = -5$
Fourth factor:
$y - 2 = 0$
Add $2$ to each side of the equation:
$y = 2$
Restrictions: $y \ne -5, 0, 2$
