Exercise 69 Page 644

We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors. We simplified the given expression. Finally, we will identify the restrictions on the variables from the denominator of the simplified expression and from any other denominator used. For simplicity, we will use their factored forms.

1/2x-3x^4

FactorOut

Factor out x

1/x(2-3x^3)

Let's do it!

Denominator	Restrictions on the Denominator	Restrictions on the Variable
3x^2(2-3x^3)	x ≠ 0, 2-3x^3 ≠ 0	x ≠ 0, x ≠ sqrt(2/3)
x(2-3x^3)	x ≠ 0, 2-3x^3 ≠ 0	x ≠ 0, x ≠ sqrt(2/3)

We found two unique restrictions on the variable. x ≠ 0, x ≠ sqrt(2/3)