If [tex]$u(x)=x^5-x^4+x^2$[/tex] and [tex]$v(x)=-x^2$[/tex], which expression is equivalent to [tex]\left(\frac{u}{v}\right)(x)[/tex]?

A. [tex]x^3-x^2[/tex]
B. [tex]-x^3+x^2[/tex]
C. [tex]-x^3+x^2-1[/tex]
D. [tex]x^3-x^2+1[/tex]



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]