Question

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

6𝑥^2−20𝑥+16/20−8𝑥−𝑥2

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify the expression 6𝑥^2−20𝑥+16/20−8𝑥−𝑥^2, we can factor both the numerator and the denominator.

Numerator:
We can factor out a 2 from each term: 2(3𝑥^2 - 10𝑥 + 8).
Now we can factor the quadratic expression: 2(3𝑥 - 4)(𝑥 - 2).

Denominator:
We can rearrange the terms: -𝑥^2 - 8𝑥 + 20.
Now we can also factor the quadratic expression: -(𝑥 - 2)(𝑥 + 10).

Now we can simplify the expression by canceling out common factors:
(2(3𝑥 - 4)(𝑥 - 2)) / (-(𝑥 - 2)(𝑥 + 10)).

At this point, we can see that (𝑥 - 2) appears in both the numerator and the denominator, so we can cancel it out:
2(3𝑥 - 4) / -(𝑥 + 10).

Therefore, the simplified form of the rational expression is 2(3𝑥 - 4) / -(𝑥 + 10).

Restrictions on the variables:
The expression is defined for all real numbers except for the values that make the denominator zero, which are x = -10 and x = 2. Therefore, the restrictions on the variable are: x ≠ -10 and x ≠ 2.