Question

Add the rational expressions to find the sum: −4x−4+x2−2x2x−8
. Put the answer into its simplest form.(1 point)
Responses

x2−2x−82x−8
Start Fraction x squared minus 2 x minus 8 over 2 x minus 8 End Fraction

x2−2x2x
Start Fraction x squared minus 2 x over 2 x End Fraction

x+22
Start Fraction x plus 2 over 2 End Fraction

x2−2x−4x−4

Answer 1

To add the rational expressions \(-\frac{4}{x-8} + \frac{x^2 - 2}{2x}\), we first need a common denominator. The least common denominator (LCD) of \(x - 8\) and \(2x\) is \(2x(x - 8)\).

Now we will rewrite each fraction to have the common denominator:

1. For \(-\frac{4}{x - 8}\):
\[
-\frac{4}{x - 8} = - \frac{4 \cdot 2x}{(x - 8) \cdot 2x} = -\frac{8x}{2x(x - 8)}
\]

2. For \(\frac{x^2 - 2}{2x}\):
\[
\frac{x^2 - 2}{2x} = \frac{(x^2 - 2)(x - 8)}{2x(x - 8)} = \frac{x^3 - 8x^2 - 2x + 16}{2x(x - 8)}
\]

Now we can combine the two fractions:
\[
-\frac{8x}{2x(x - 8)} + \frac{x^3 - 8x^2 - 2x + 16}{2x(x - 8)}
\]
This simplifies to:
\[
\frac{-8x + x^3 - 8x^2 - 2x + 16}{2x(x - 8)} = \frac{x^3 - 8x^2 - 10x + 16}{2x(x - 8)}
\]

Now, we'll factor the numerator \(x^3 - 8x^2 - 10x + 16\). Unfortunately, it doesn't factor nicely, so we'll express our answer as:
\[
\frac{x^3 - 8x^2 - 10x + 16}{2x(x - 8)}
\]

Let us check if the numerator can be simplified. The final expression would be the answer in its simplest form. Since factoring does not yield simple results and we don't see a way to reduce further, we have:

\[
\frac{x^3 - 8x^2 - 10x + 16}{2x(x - 8)}
\]

Looking at the options provided:

1. \( \frac{x^2 - 2x - 8}{2x - 8} \)
2. \( \frac{x^2 - 2x}{2x} \)
3. \( \frac{x + 2}{2} \)
4. \( x^2 - 2x - 4x - 4 \)

None exactly match the result we got, so none of the choices are correct based on simplification.

In conclusion, the answer to \(-\frac{4}{x-8} + \frac{x^2 - 2}{2x}\) in its simplest form is:

\[
\frac{x^3 - 8x^2 - 10x + 16}{2x(x - 8)}
\]