To find which expression is equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] given that [tex]\(u(x) = x^5 - x^4 + x^2\)[/tex] and [tex]\(v(x) = -x^2\)[/tex], let's perform the following steps:
1. Identify the expressions for [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]:
- [tex]\(u(x) = x^5 - x^4 + x^2\)[/tex]
- [tex]\(v(x) = -x^2\)[/tex]
2. Form the quotient [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex]:
[tex]\[
\left(\frac{u}{v}\right)(x) = \frac{x^5 - x^4 + x^2}{-x^2}
\][/tex]
3. Divide each term in the numerator by the term in the denominator [tex]\(-x^2\)[/tex]:
[tex]\[
\left(\frac{u}{v}\right)(x) = \left(\frac{x^5}{-x^2}\right) + \left(\frac{-x^4}{-x^2}\right) + \left(\frac{x^2}{-x^2}\right)
\][/tex]
4. Simplify each term individually:
[tex]\[
\frac{x^5}{-x^2} = -x^{5-2} = -x^3
\][/tex]
[tex]\[
\frac{-x^4}{-x^2} = x^{4-2} = x^2
\][/tex]
[tex]\[
\frac{x^2}{-x^2} = -1
\][/tex]
5. Combine these simplified terms:
[tex]\[
-x^3 + x^2 - 1
\][/tex]
So, the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] is [tex]\( -x^3 + x^2 - 1 \)[/tex].
Therefore, the correct option is:
[tex]\[
-x^3 + x^2 - 1
\][/tex]
