Question

simplify the rational expression.\\(
\frac{x^{3} - 216}{3x - 18} = \square\\)
(use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Factor the numerator and denominator

First, factor the numerator \(x^3 - 216\) using the difference of cubes formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). Here, \(a = x\) and \(b = 6\) (since \(6^3=216\)), so \(x^3 - 216=(x - 6)(x^2+6x + 36)\).

Next, factor the denominator \(3x - 18\) by taking out the common factor of 3: \(3x - 18 = 3(x - 6)\).

So the rational expression becomes \(\frac{(x - 6)(x^2+6x + 36)}{3(x - 6)}\).

Step2: Cancel out the common factor

We can cancel out the common factor \((x - 6)\) from the numerator and the denominator (assuming \(x
eq6\) to avoid division by zero). After canceling, we are left with \(\frac{x^2+6x + 36}{3}\).

Answer:

\(\frac{x^2 + 6x + 36}{3}\)