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

The numerator can be factored as:
6𝑥^2−20𝑥+16 = 2(3𝑥^2−10𝑥+8) = 2(𝑥−2)(3𝑥−4)

The denominator can be factored as:
20−8𝑥−𝑥^2 = (𝑥+4)(−𝑥+5)

Now we can rewrite the rational expression as:
(2(𝑥−2)(3𝑥−4)) / ((𝑥+4)(−𝑥+5))

The expression is already in simplest form. The restrictions on the variables occur when the denominator is equal to zero. So, we set each factor of the denominator equal to zero and solve for 𝑥:

𝑥+4 = 0 -> 𝑥 = -4
−𝑥+5 = 0 -> 𝑥 = 5

Therefore, the restrictions on the variable 𝑥 are 𝑥 ≠ -4 and 𝑥 ≠ 5.