Question

8-1: a rational expression with non - permissible value of ( x = - 2,0 ) has been simplified to ( \frac{x}{3} ). determine an equivalent rational expression for the simplified expression.

Explanation:

Step1: Identify excluded values

Non-permissible values: $x=-2, 0$

Step2: Multiply numerator/denominator by common factor

To introduce the excluded values, multiply numerator and denominator by $x(x+2)$:
$\frac{x}{3} \times \frac{x(x+2)}{x(x+2)}$

Step3: Expand the numerator

Multiply terms in the numerator:
$\frac{x \cdot x(x+2)}{3x(x+2)} = \frac{x^2(x+2)}{3x(x+2)}$

Step4: Simplify expanded numerator

Distribute $x^2$ in the numerator:
$\frac{x^3 + 2x^2}{3x(x+2)}$ (or $\frac{x^3 + 2x^2}{3x^2 + 6x}$ after expanding denominator)

Answer:

$\boldsymbol{\frac{x^3 + 2x^2}{3x^2 + 6x}}$ (or equivalent form $\boldsymbol{\frac{x^2(x+2)}{3x(x+2)}}$)