If $$ u(x)=x^{5}-x^{4}+x^{2} $$ and $$ v(x)=-x^{2} $$, which expression is equivalent to $$ \left(\frac{u}{v}\right)(x) $$ ? $$ x^{3}-x^{2} $$ $$ -x^{3}+x^{2} $$ $$ -x^{3}+x^{2}-1 $$ $$ x^{3}-x^{2}+1 $$

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UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Divide the terms:
  $$\frac{x^{5}-x^{4}+x^{2}}{-x^{2}}$$
- step1: Rewrite the fraction:
  $$-\frac{x^{5}-x^{4}+x^{2}}{x^{2}}$$
- step2: Factor:
  $$-\frac{x^{2}\left(x^{3}-x^{2}+1\right)}{x^{2}}$$
- step3: Reduce the fraction:
  $$-\left(x^{3}-x^{2}+1\right)$$
- step4: Calculate:
  $$-x^{3}+x^{2}-1$$
To find the expression equivalent to $$ \left(\frac{u}{v}\right)(x) $$, we start with the given functions:

1. $$ u(x) = x^{5} - x^{4} + x^{2} $$
2. $$ v(x) = -x^{2} $$

Now, we compute $$ \frac{u(x)}{v(x)} $$:

$$
\frac{u(x)}{v(x)} = \frac{x^{5} - x^{4} + x^{2}}{-x^{2}}
$$

Next, we simplify this expression:

$$
\frac{x^{5}}{-x^{2}} + \frac{-x^{4}}{-x^{2}} + \frac{x^{2}}{-x^{2}} = -x^{3} + x^{2} - 1
$$

Thus, the expression equivalent to $$ \left(\frac{u}{v}\right)(x) $$ is:

$$
-x^{3} + x^{2} - 1
$$

Among the provided options, the correct answer is:

$$-x^{3} + x^{2} - 1$$
The expression equivalent to $$ \left(\frac{u}{v}\right)(x) $$ is $$ -x^{3} + x^{2} - 1 $$.

If \( u(x)=x^{5}-x^{4}+x^{2} \) and \( v(x)=-x^{2} \), which expression is equivalent to \( \left(\frac{u}{v}\right)(x) \) ? \( x^{3}-x^{2} \) \( -x^{3}+x^{2} \) \( -x^{3}+x^{2}-1 \) \( x^{3}-x^{2}+1 \)

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