Perform the following operation and express in simplest form. start fraction, x, squared, minus, 4, divided by, x, minus, 2, end fraction, divided by, start fraction, x, squared, minus, 2, x, minus, 8, divided by, 3, x, cubed, end fraction x−2 x 2 −4 ÷ 3x 3 x 2 −2x−8
To perform the given operation and express in the simplest form, we first need to simplify each fraction individually.
Starting with the numerator of the first fraction: x^2 - 4
This expression is the difference of squares, which can be factored as (x + 2)(x - 2).
Moving on to the numerator of the second fraction: x^2 - 2x - 8
This expression can be factored as (x - 4)(x + 2).
Now we can rewrite the original expression in factored form:
[(x + 2)(x - 2)] / [(x - 4)(x + 2)] ÷ (3x^3)
The numerator and denominator of the main fraction have a common factor of (x + 2), which can be canceled out. Additionally, the (x - 2) term in the numerator and the (x - 4) term in the denominator can be canceled out:
[(x + 2)(x - 2)] / [(x - 4)(x + 2)] ÷ (3x^3) = 1 / (3x^3)
Therefore, the simplified form of the given expression is 1 / (3x^3).