Question

Write each rational expression in simplest form. State all restrictions on the variables.

2𝑥^2−10𝑥−48/18−2𝑥^2

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify the rational expression 2𝑥^2−10𝑥−48/18−2𝑥^2, we can start by factoring the numerator and denominator.

The numerator factors to:
2𝑥^2−10𝑥−48 = (2𝑥+6)(𝑥−8)

The denominator rearranges to:
18−2𝑥^2 = -2(𝑥^2-9) = -2(𝑥+3)(𝑥-3)

Now we can simplify the expression:
(2𝑥+6)(𝑥−8)/-2(𝑥+3)(𝑥-3)

The (2𝑥+6) term can be reduced by factoring out a 2:
2(𝑥+3)(𝑥-8)/-2(𝑥+3)(𝑥-3)

The 2's cancel out:
(𝑥+3)(𝑥-8)/-(𝑥+3)(𝑥-3)

Finally, the (𝑥+3) terms can also cancel out:
-(𝑥-8)/(𝑥-3)

The simplified rational expression is -(𝑥-8)/(𝑥-3).
The restriction on the variable is that 𝑥 cannot equal 3 or 8, since those values would result in division by zero.