Question

Subtract the rational expressions
2x−12
x
2

​
−
x−6
18
​

Simplify the answer.

Answer 1

To subtract the rational expressions \( \frac{2x - 12}{x^2} \) and \( \frac{x - 6}{18} \), we first need to find a common denominator.

The common denominator for \( x^2 \) and \( 18 \) is \( 18x^2 \).

Now, we will rewrite each fraction with the common denominator:

1. For \( \frac{2x - 12}{x^2} \):
\[
\frac{2x - 12}{x^2} = \frac{(2x - 12) \cdot 18}{x^2 \cdot 18} = \frac{36x - 216}{18x^2}
\]

2. For \( \frac{x - 6}{18} \):
\[
\frac{x - 6}{18} = \frac{(x - 6) \cdot x^2}{18 \cdot x^2} = \frac{x^2(x - 6)}{18x^2} = \frac{x^3 - 6x^2}{18x^2}
\]

Now we can subtract the two expressions:
\[
\frac{36x - 216}{18x^2} - \frac{x^3 - 6x^2}{18x^2} = \frac{(36x - 216) - (x^3 - 6x^2)}{18x^2}
\]

Next, simplify the numerator:
\[
(36x - 216) - (x^3 - 6x^2) = 36x - 216 - x^3 + 6x^2 = -x^3 + 6x^2 + 36x - 216
\]

Now, we can write the final expression:
\[
\frac{-x^3 + 6x^2 + 36x - 216}{18x^2}
\]

To simplify further, we can factor the numerator. Let's factor out \(-1\):
\[
-1(x^3 - 6x^2 - 36x + 216)
\]

Now we will check if the cubic polynomial \( x^3 - 6x^2 - 36x + 216 \) can be factored. By using the Rational Root Theorem and synthetic division, we can find that \( x = 6 \) is a root. Thus, we can factor it as follows:
\[
x^3 - 6x^2 - 36x + 216 = (x - 6)(x^2 - 36) = (x - 6)(x - 6)(x + 6) = (x - 6)^2(x + 6)
\]

So the complete factorization gives:
\[
-x^3 + 6x^2 + 36x - 216 = -(x - 6)^2(x + 6)
\]

Substituting this back into our expression, we have:
\[
\frac{-(x - 6)^2(x + 6)}{18x^2}
\]

We can finally write the result as:
\[
\frac{-(x - 6)^2(x + 6)}{18x^2}
\]

Thus, the simplified expression after subtracting the two rational functions is:
\[
\frac{-(x - 6)^2(x + 6)}{18x^2}
\]