To determine which expression is equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] given the functions [tex]\(u(x) = x^5 - x^4 + x^2\)[/tex] and [tex]\(v(x) = -x^2\)[/tex], we need to compute the quotient [tex]\(\frac{u(x)}{v(x)}\)[/tex].
First, write down the given functions:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
Now, compute the quotient [tex]\(\frac{u(x)}{v(x)}\)[/tex]:
[tex]\[ \frac{u(x)}{v(x)} = \frac{x^5 - x^4 + x^2}{-x^2} \][/tex]
We can simplify this expression by dividing each term in the numerator by [tex]\( -x^2 \)[/tex]:
[tex]\[ \frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2} \][/tex]
Simplify each term:
[tex]\[ \frac{x^5}{-x^2} = -x^3 \][/tex]
[tex]\[ \frac{x^4}{-x^2} = -x^2 \][/tex]
[tex]\[ \frac{x^2}{-x^2} = -1 \][/tex]
Putting it all together, the simplified expression is:
[tex]\[ \frac{x^5 - x^4 + x^2}{-x^2} = -x^3 + x^2 - 1 \][/tex]
Thus, the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] is:
[tex]\[ -x^3 + x^2 - 1 \][/tex]
Therefore, the correct choice from the provided options is:
[tex]\[ -x^3 + x^2 - 1 \][/tex]
