Question

Find the difference of the following rational expressions.(1 point)
x2−2x2−5−4x+15−x2=
$$

Answer 1

To find the difference of the given rational expressions, we need to simplify the expression step by step.

The given expression is:

\[
\frac{x^2 - 2}{x^2 - 5} - \frac{4x + 15}{-x^2}
\]

First, let's simplify the second term \(\frac{4x + 15}{-x^2}\):

\[
\frac{4x + 15}{-x^2} = -\frac{4x + 15}{x^2}
\]

Now we can rewrite the expression:

\[
\frac{x^2 - 2}{x^2 - 5} + \frac{4x + 15}{x^2}
\]

Next, we need a common denominator to combine these two fractions. The common denominator will be:

\[
(x^2 - 5)x^2
\]

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

1. For the first fraction:
\[
\frac{x^2 - 2}{x^2 - 5} = \frac{(x^2 - 2)x^2}{(x^2 - 5)x^2} = \frac{x^4 - 2x^2}{(x^2 - 5)x^2}
\]

2. For the second fraction:
\[
\frac{4x + 15}{x^2} = \frac{(4x + 15)(x^2 - 5)}{(x^2)(x^2 - 5)} = \frac{(4x + 15)(x^2 - 5)}{(x^2 - 5)x^2}
\]

Now, we can combine the fractions:

\[
\frac{x^4 - 2x^2 + (4x + 15)(x^2 - 5)}{(x^2 - 5)x^2}
\]

Next, we need to expand \((4x + 15)(x^2 - 5)\):

\[
(4x + 15)(x^2 - 5) = 4x \cdot x^2 + 15 \cdot x^2 - 4x \cdot 5 - 15 \cdot 5 = 4x^3 + 15x^2 - 20x - 75
\]

Now, substitute this back into our combined expression:

\[
\frac{x^4 - 2x^2 + 4x^3 + 15x^2 - 20x - 75}{(x^2 - 5)x^2}
\]

Combine the like terms in the numerator:

\[
x^4 + 4x^3 + (-2x^2 + 15x^2) - 20x - 75 = x^4 + 4x^3 + 13x^2 - 20x - 75
\]

Thus, the final expression is:

\[
\frac{x^4 + 4x^3 + 13x^2 - 20x - 75}{(x^2 - 5)x^2}
\]

This is the simplified result of the difference of the rational expressions.