Question

simplify the following rational expressions keep the denominator in its factored form

x2^ x blank
(x-2) (x+5) - x-2 = blank

Answer 1

To simplify the given rational expression:

\[
\frac{x^2}{(x - 2)(x + 5)} - \frac{x - 2}{1}
\]

First, let's rewrite the second term with a common denominator. The denominator of the first term is \((x - 2)(x + 5)\), so we need to give the second term the same denominator:

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

Now, let's distribute this out:

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

Expanding \((x - 2)(x + 5)\):

\[
(x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10
\]

Now, substituting:

\[
\frac{x^2 - (x^2 + 3x - 10)}{(x - 2)(x + 5)} = \frac{x^2 - x^2 - 3x + 10}{(x - 2)(x + 5)} = \frac{-3x + 10}{(x - 2)(x + 5)}
\]

Thus, the expression simplified is:

\[
\frac{10 - 3x}{(x - 2)(x + 5)}
\]

This is our final simplified expression, with the denominator kept in its factored form.