Answer
$\dfrac{x(x - 1)}{3(x + 1)}$
Restrictions: $x \ne -1, 0, 1$
Work Step by Step
Factor each polynomial:
$\dfrac{x^2}{(x + 1)(x + 1)} \div \dfrac{3x}{(x - 1)(x + 1)}$
To divide one rational expression by another, multiply the first expression by the reciprocal of the second expression:
$=\dfrac{x^2}{(x + 1)(x + 1)} \cdot \dfrac{(x - 1)(x + 1)}{3x}$
Rewrite as one expression:
$=\dfrac{x^2 \cdot (x - 1)(x + 1)}{(x + 1)(x + 1) \cdot 3x}$
Cancel out common factorsin the numerator and denominator:
$=\dfrac{x(x - 1)}{3(x + 1)}$
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:
$x + 1 = 0$
Subtract $1$ from each side of the equation:
$x = -1$
Second factor:
$x - 1 = 0$
Add $1$ to each side of the equation:
$x = 1$
Third factor:
$3x = 0$
Divide both sides of the equation by $3$:
$x = 0$
Restrictions: $x \ne -1, 0, 1$
