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 :

Sure, let's solve this step by step.

Given the functions:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]

We need to find the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex].

First, we write the fraction [tex]\(\frac{u(x)}{v(x)}\)[/tex]:
[tex]\[ \frac{u(x)}{v(x)} = \frac{x^5 - x^4 + x^2}{-x^2} \][/tex]

Next, we simplify this expression by dividing each term in the numerator by the denominator [tex]\(-x^2\)[/tex]:

1. Divide [tex]\(x^5\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^5}{-x^2} = -x^{5-2} = -x^3 \][/tex]

2. Divide [tex]\(x^4\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^4}{-x^2} = -x^{4-2} = -x^2 \][/tex]

3. Divide [tex]\(x^2\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[ \frac{x^2}{-x^2} = -1 \][/tex]

Combining these results, we get:
[tex]\[ \frac{x^5 - x^4 + x^2}{-x^2} = -x^3 + x^2 - 1 \][/tex]

Therefore, the expression equivalent to [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] is:
[tex]\[ -x^3 + x^2 - 1 \][/tex]

Among the given options, the correct one is:
[tex]\[ \boxed{-x^3 + x^2 - 1} \][/tex]