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

Question

48𝑎−3𝑎^3/2𝑎^2+2𝑎−8𝑎

Answer 1

To simplify the expression, we need to factor out a common factor from the numerator and denominator.

48𝑎−3𝑎^3/2𝑎^2+2𝑎−8𝑎

First, we notice that we can factor out a common factor of 𝑎 from the numerator and denominator:

𝑎(48−3𝑎^2)/𝑎(2𝑎+2−8)

Next, we can simplify and cancel out the common factors of 𝑎:

(48−3𝑎^2)/(2𝑎−6)

Now, we can further simplify by factoring out the common factor of 3 from the numerator:

(3(16−𝑎^2))/(2𝑎−6)

The numerator can be simplified even further by factoring 𝑎^2−16 as the difference of squares:

(3(4+𝑎)(4−𝑎))/(2𝑎−6)

Finally, we can simplify the expression:

(3(4+𝑎)(4−𝑎))/(2(𝑎−3))

The restrictions on the variables are 𝑎 ≠ 0 and 𝑎 ≠ 3.