Answer :
To solve the problem, we need to determine which expression is equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] where [tex]\( u(x) = x^5 - x^4 + x^2 \)[/tex] and [tex]\( v(x) = -x^2 \)[/tex].
First, we will substitute the expressions for [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] into [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]
Next, we will simplify the rational expression by dividing each term in the numerator by the term in the denominator:
[tex]\[ \left(\frac{x^5 - x^4 + x^2}{-x^2}\right) = \frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2} \][/tex]
Simplifying each term separately, we get:
[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]
Combining these terms, we obtain:
[tex]\[ \left(\frac{u}{v}\right)(x) = -x^3 - x^2 - 1 \][/tex]
From the provided options, the equivalent expression we derived matches the third option:
[tex]\[ - x^3 + x^2 - 1 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-x^3 + x^2 - 1} \][/tex]
First, we will substitute the expressions for [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] into [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]
Next, we will simplify the rational expression by dividing each term in the numerator by the term in the denominator:
[tex]\[ \left(\frac{x^5 - x^4 + x^2}{-x^2}\right) = \frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2} \][/tex]
Simplifying each term separately, we get:
[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]
Combining these terms, we obtain:
[tex]\[ \left(\frac{u}{v}\right)(x) = -x^3 - x^2 - 1 \][/tex]
From the provided options, the equivalent expression we derived matches the third option:
[tex]\[ - x^3 + x^2 - 1 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-x^3 + x^2 - 1} \][/tex]
