Simplify each rational expression. State all restrictions on the variables
𝑎^2−4 / 2𝑎+1 × 6𝑎+3 / 𝑎−2
To simplify the rational expression, we can cancel out common factors in the numerator and denominator.
First, let's factorize the numerator and denominator individually:
Numerator:
𝑎^2−4 = (𝑎+2)(𝑎−2)
Denominator:
2𝑎+1 = (2𝑎+1)
𝑎−2 = (𝑎−2)
Now, let's cancel out common factors:
𝑎^2−4/(2𝑎+1) × 6𝑎+3/(𝑎−2) = (𝑎+2)(𝑎−2)/(2𝑎+1) × 6(𝑎+1)(𝑎−3)/(𝑎−2)
The (𝑎−2) factor in the numerator cancels out with the (𝑎−2) factor in the denominator. Also, the (𝑎+1) factor in the numerator cancels out with the (𝑎+1) factor in the denominator.
Therefore, the simplified expression is:
(𝑎+2)6(𝑎−3)/(2𝑎+1)
The restrictions on the variable are that 𝑎 cannot equal 2 and 𝑎 cannot equal -1 since they would result in division by zero.