Question

Write each rational expression in simplest form. State all restrictions on the variables.
48𝑎−3𝑎^3/2𝑎^2+2𝑎−8a

Answer 1

To simplify the rational expression 48𝑎−3𝑎^3/2𝑎^2+2𝑎−8a, we can factor out the greatest common factor in the numerator, which is 3𝑎:

3𝑎(16−𝑎^2)/2𝑎^2+2𝑎−8a

Next, we can factor out the greatest common factor in the denominator, which is 2𝑎:

3𝑎(16−𝑎^2)/2𝑎(𝑎+1)−4(𝑎+1)

Now, we can cancel out the common factors (3𝑎)/(2𝑎):

(16−𝑎^2)/(𝑎+1)−2(𝑎+1)

The simplified form of the rational expression is (16−𝑎^2)/(𝑎+1)−2(𝑎+1).

Now, let's state the restrictions on the variables. In the original expression, the denominator 2𝑎^2+2𝑎−8a cannot be equal to zero since division by zero is undefined. Therefore, we can solve the quadratic equation 2𝑎^2+2𝑎−8a=0 to find the restrictions.

Using the quadratic formula, we get:

𝑎 = (-2 ± √(2^2-4(2)(-8a)))/(2(2))

Simplifying the equation:

𝑎 = (-2 ± √(4+64a))/(4)

The restrictions on the variable 𝑎 are the values that make the denominator of the original expression zero. Thus, the restrictions are 𝑎 = (-2 + √(4+64a))/(4) and 𝑎 = (-2 - √(4+64a))/(4).