Answer :
To solve the problem, let's start by analyzing and simplifying the given functions [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]:
1. The functions are given as:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
2. We need to find the expression [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex], which means we need to divide [tex]\(u(x)\)[/tex] by [tex]\(v(x)\)[/tex]:
[tex]\[ \frac{u(x)}{v(x)} = \frac{x^5 - x^4 + x^2}{-x^2} \][/tex]
3. We can separate and simplify each term in the numerator by dividing by the denominator:
[tex]\[ \frac{x^5 - x^4 + x^2}{-x^2} = \frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2} \][/tex]
4. Simplify each term:
[tex]\[ \frac{x^5}{-x^2} = -x^3, \quad \frac{x^4}{-x^2} = -x^2, \quad \frac{x^2}{-x^2} = -1 \][/tex]
5. Combine the simplified terms:
[tex]\[ -x^3 - (-x^2) - 1 = -x^3 + x^2 - 1 \][/tex]
Thus, the expression [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] simplifies to:
[tex]\[ -x^3 + x^2 - 1 \][/tex]
The equivalent expression is [tex]\(-x^3 + x^2 - 1\)[/tex]. Hence, among the given options, the correct one is:
[tex]\(\boxed{-x^3 + x^2 - 1}\)[/tex]
1. The functions are given as:
[tex]\[ u(x) = x^5 - x^4 + x^2 \][/tex]
[tex]\[ v(x) = -x^2 \][/tex]
2. We need to find the expression [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex], which means we need to divide [tex]\(u(x)\)[/tex] by [tex]\(v(x)\)[/tex]:
[tex]\[ \frac{u(x)}{v(x)} = \frac{x^5 - x^4 + x^2}{-x^2} \][/tex]
3. We can separate and simplify each term in the numerator by dividing by the denominator:
[tex]\[ \frac{x^5 - x^4 + x^2}{-x^2} = \frac{x^5}{-x^2} - \frac{x^4}{-x^2} + \frac{x^2}{-x^2} \][/tex]
4. Simplify each term:
[tex]\[ \frac{x^5}{-x^2} = -x^3, \quad \frac{x^4}{-x^2} = -x^2, \quad \frac{x^2}{-x^2} = -1 \][/tex]
5. Combine the simplified terms:
[tex]\[ -x^3 - (-x^2) - 1 = -x^3 + x^2 - 1 \][/tex]
Thus, the expression [tex]\(\left(\frac{u}{v}\right)(x)\)[/tex] simplifies to:
[tex]\[ -x^3 + x^2 - 1 \][/tex]
The equivalent expression is [tex]\(-x^3 + x^2 - 1\)[/tex]. Hence, among the given options, the correct one is:
[tex]\(\boxed{-x^3 + x^2 - 1}\)[/tex]
