Question

How to solve this problem

Answer 1

First, since we are subtracting fractions, we are going to want to find a common denominator between the terms we are subtracting. In this case, $(x + 2)$, $(x - 2)$, and $x^2$ are very different terms, meaning that we are going to have to multiply both fractions by the terms they are missing in the denominators.

The term $\dfrac{3x}{(x + 2)(x - 2)}$ does not have an $x^2$ in the denominator, meaning that we are going to need to multiply both the numerator and the denominator of the fraction by $x^2$. We have to multiply it by both the numerator and the denominator to keep the fraction similar to its prior form. Doing this results in:

$\Bigg(\dfrac{3x}{(x + 2)(x - 2)}\Bigg)\Bigg(\dfrac{x^2}{x^2}\Bigg)$

$\dfrac{3x^3}{(x + 2)(x - 2)(x^2)}$

In the second term which is being subtracted, the denominator is absent of the $(x + 2)$ and $(x - 2)$ terms, meaning that we will have to multiply both the numerators and denominators of the fraction by these terms to give the second fraction a like denominator:

$\Bigg(\dfrac{1}{x^2}\Bigg)\Bigg(\dfrac{(x + 2)(x - 2)}{(x + 2)(x - 2)}\Bigg)$

$\dfrac{(x + 2)(x - 2)}{(x + 2)(x - 2)(x^2)}$

Using these terms, our subtraction problem looks like this:

$\dfrac{3x^3}{(x + 2)(x - 2)(x^2)} - \dfrac{(x + 2)(x - 2)}{(x + 2)(x - 2)(x^2)}$

We can now use our common denominator to simplify this problem to just one fraction:

$\dfrac{3x^3 - (x + 2)(x - 2)}{(x + 2)(x - 2)(x^2)}$

Now, using our algebraic operations, we can simplify this fraction into something more manageable:

$\dfrac{3x^3 - (x + 2)(x - 2)}{(x + 2)(x - 2)(x^2)}$

$\dfrac{3x^3 - (x^2 - 4)}{(x^2 - 4)(x^2)}$

$\dfrac{3x^3 - x^2 + 4}{x^4 - 4x^2}$

Our simplified expression would be:

$\boxed{\dfrac{3x^3 - x^2 + 4}{x^4 - 4x^2}}$

(Keep in mind that this is the same exact expression, represented in a different way. This means that there are many, many ways to represent the original expression, but this one is one that I feel very well simplifies the original problem.)